JEE Applications of Derivatives Questions

To find the number of points of local minima $$m$$ and local maxima $$n$$, we analyze the behavior of the absolute value function by breaking it down into intervals based on its critical points.

The critical points inside the absolute value expressions are $$x = -2$$ and $$x = 0$$.

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Step 1: Simplify the inner expression $$g(x) = |x + 2| - 2|x|$$ across different intervals

Case 1: For $$x < -2$$

Both expressions inside the absolute values are negative:

$$g(x) = -(x + 2) - 2(-x) = -x - 2 + 2x = x - 2$$

Case 2: For $$-2 \le x < 0$$

The first expression is non-negative and the second is negative:

$$g(x) = (x + 2) - 2(-x) = x + 2 + 2x = 3x + 2$$

Case 3: For $$x \ge 0$$

Both expressions inside the absolute values are non-negative:

$$g(x) = (x + 2) - 2(x) = x + 2 - 2x = -x + 2$$

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Step 2: Find the roots of the inner expression to account for the outer absolute value

The function is defined as $$f(x) = |g(x)|$$. The outer absolute value will reflect any negative portions of $$g(x)$$ above the x-axis, creating sharp corners (local minima where $$f(x) = 0$$) at its roots.

Checking for roots (g(x) = 0) in each interval:

In Case 1 ($$x < -2$$): $$x - 2 = 0 \implies x = 2$$ (Not in interval).

In Case 2 ($$-2 \le x < 0$$): $$3x + 2 = 0 \implies x = -\frac{2}{3}$$ (Valid root).

In Case 3 ($$x \ge 0$$): $$-x + 2 = 0 \implies x = 2$$ (Valid root).

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Step 3: Analyze the turning points and identify local extrema

We combine the structural boundaries ($$x = -2, 0$$) and the reflection roots ($$x = -\frac{2}{3}, 2$$) to evaluate the behavior of $$f(x)$$:

At $$x = -2$$: $$f(-2) = |(-2) - 2| = |-4| = 4$$

As $$x \to -\infty$$: $$g(x) = x - 2$$ is negative, so $$f(x) = |x - 2| = -x + 2$$. As $$x$$ decreases, $$f(x)$$ increases. Thus, moving from left to right, $$f(x)$$ decreases down to 4 at $$x = -2$$.

Between $$x = -2$$ and $$x = -\frac{2}{3}$$: $$g(x) = 3x + 2$$ goes from $$-4$$ up to $$0$$. Thus, $$f(x)$$ decreases from 4 to 0. This makes $$x = -2$$ a turning point where the function changes direction from decreasing to increasing if we look at $$g(x)$$, but because of the absolute value, it continues decreasing.

Let's look at the exact values sequentially from left to right:

1. From $$-\infty$$ to $$x = -2$$, the function decreases from $$\infty$$ to 4.

2. From $$x = -2$$ to $$x = -\frac{2}{3}$$, the function decreases from 4 to 0.

3. At $$x = -\frac{2}{3}$$, $$f\left(-\frac{2}{3}\right) = 0$$. Since $$f(x) \ge 0$$ everywhere, this point is a local minimum.

4. From $$x = -\frac{2}{3}$$ to $$x = 0$$, the function increases from 0 to $$f(0) = |2| = 2$$.

5. From $$x = 0$$ to $$x = 2$$, the function decreases from 2 to $$f(2) = 0$$. This makes $$x = 0$$ a local maximum.

6. At $$x = 2$$, $$f(2) = 0$$. Since $$f(x) \ge 0$$ everywhere, this point is another local minimum.

7. For $$x > 2$$, $$g(x) = -x + 2$$ becomes increasingly negative, so $$f(x) = x - 2$$ increases towards $$\infty$$.

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Step 4: Count the extrema points

Local minima occur at the points where the function reaches 0 and turns upward:

$$x = -\frac{2}{3}$$ and $$x = 2$$

Thus, the number of points of local minima is $$m = 2$$.

Local maxima occur at the peak between these valleys:

$$x = 0$$

Thus, the number of points of local maxima is $$n = 1$$.

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Evaluating the final required sum:

$$m + n = 2 + 1 = 3$$

Therefore, the value of $$m + n$$ is equal to 3.

A spherical chocolate ball has an ice-cream layer of uniform thickness. When the thickness is 1 cm, the ice-cream melts at 81 cm³/min and the thickness decreases at $$\frac{1}{4\pi}$$ cm/min.

Set up variables

Let r = radius of the chocolate ball and t = thickness of ice-cream layer.

The outer radius is R = r + t.

Volume of ice-cream: $$V = \frac{4}{3}\pi(r+t)^3 - \frac{4}{3}\pi r^3$$

Find the rate of change of volume

$$\frac{dV}{dt_{time}} = 4\pi(r+t)^2 \cdot \frac{d(r+t)}{dt_{time}} = 4\pi(r+t)^2 \cdot \frac{dt}{dt_{time}}$$

(since r is constant, $$\frac{d(r+t)}{dt_{time}} = \frac{dt}{dt_{time}}$$)

Note: The volume is decreasing, so $$\frac{dV}{dt_{time}} = -81$$ cm³/min

And the thickness decreases, so $$\frac{dt}{dt_{time}} = -\frac{1}{4\pi}$$ cm/min

Substitute values when t = 1 cm

$$-81 = 4\pi(r+1)^2 \times \left(-\frac{1}{4\pi}\right)$$

$$-81 = -(r+1)^2$$

$$(r+1)^2 = 81$$

$$r+1 = 9$$

$$r = 8$$ cm

Find the surface area of the chocolate ball

$$S = 4\pi r^2 = 4\pi(8)^2 = 256\pi \text{ cm}^2$$

The correct answer is Option 2: $$256\pi$$.

We have two functions: $$f(x)=\log_e(x^2+2x+4)$$ for all $$x\in\mathbb{R}$$ and $$g(x)=\dfrac{4}{1+e^{-2x}}$$ whose range is $$(0,\,4)$$.

The composite function is$$(f\circ g^{-1})(y)=f\!\bigl(g^{-1}(y)\bigr),\qquad 0\lt y\lt 4.$$We must find its derivative at $$y=2$$.

Step 1: Find the inverse $$g^{-1}$$.
Let $$y=g(x)=\dfrac{4}{1+e^{-2x}}.$$ Multiply: $$y\,(1+e^{-2x})=4\;\;\Longrightarrow\;\;y\,e^{-2x}=4-y.$$ Therefore $$e^{-2x}=\dfrac{4-y}{y}.$$ Taking natural logarithm,
$$-2x=\ln\!\Bigl(\dfrac{4-y}{y}\Bigr)\;\;\Longrightarrow\;\;x=\dfrac12\ln\!\Bigl(\dfrac{y}{4-y}\Bigr).$$ Thus$$g^{-1}(y)=\dfrac12\ln\!\Bigl(\dfrac{y}{4-y}\Bigr),\qquad 0\lt y\lt4.$$ For convenience, write$$x=g^{-1}(y)=\dfrac12\ln\!\Bigl(\dfrac{y}{4-y}\Bigr).$$

Step 2: Differentiate $$f(x)$$.
$$f(x)=\ln(x^2+2x+4)$$
Using $$\dfrac{d}{dx}\ln u=\dfrac{u'}{u}$$,
$$f'(x)=\dfrac{2x+2}{x^2+2x+4}= \dfrac{2(x+1)}{x^2+2x+4}.$$ At $$x=0$$ (value found later),
$$f'(0)=\dfrac{2(0+1)}{0^2+0+4}= \dfrac{2}{4}= \dfrac12.$$

Step 3: Differentiate $$x=g^{-1}(y)$$ with respect to $$y$$.
$$x=\dfrac12\ln\!\Bigl(\dfrac{y}{4-y}\Bigr).$$ Let $$u(y)=\dfrac{y}{4-y}.$$ Then $$x=\dfrac12\ln u.$$ First evaluate $$u'(y)$$ using the quotient rule:
$$u'(y)=\dfrac{(4-y)\cdot1-y(-1)}{(4-y)^2}= \dfrac{4}{(4-y)^2}.$$ Now$$\dfrac{d}{dy}\ln u=\dfrac{u'(y)}{u(y)}=\dfrac{4}{(4-y)^2}\,\bigg/\,\dfrac{y}{4-y}= \dfrac{4}{y(4-y)}.$$ Hence$$\dfrac{dx}{dy}= \dfrac12\cdot\dfrac{4}{y(4-y)}= \dfrac{2}{y(4-y)}.$$ At $$y=2$$,
$$\dfrac{dx}{dy}\bigg|_{y=2}= \dfrac{2}{2\,(4-2)}= \dfrac{2}{4}= \dfrac12.$$

Step 4: Value of $$x$$ when $$y=2$$.
$$x=g^{-1}(2)=\dfrac12\ln\!\Bigl(\dfrac{2}{4-2}\Bigr)=\dfrac12\ln 1=0.$$

Step 5: Apply the Chain Rule.
For the composite $$h(y)=(f\circ g^{-1})(y)=f(x)\,,$$
$$h'(y)=f'(x)\,\dfrac{dx}{dy}.$$ At $$y=2$$ (so $$x=0$$):
$$h'(2)=f'(0)\times\dfrac{dx}{dy}\bigg|_{y=2}= \dfrac12 \times \dfrac12= \dfrac14=0.25.$$

Therefore the derivative of the composite function at $$x=2$$ is $$0.25$$, which lies in the required range $$0.2-0.3$$.

$$f'(x) = \frac{2}{x-2} - 2x + a$$.

Given $$f'(x) = 0$$ at $$x=3$$ (boundary of increasing interval):

$$\frac{2}{3-2} - 2(3) + a = 0 \implies 2 - 6 + a = 0 \implies \mathbf{a = 4}$$.

Analyze $$g(x)$$ with $$a=4$$.

$$g(x) = (x-1)^3(x-2)^2$$.

$$g'(x) = 3(x-1)^2(x-2)^2 + 2(x-1)^3(x-2) = (x-1)^2(x-2)[3(x-2) + 2(x-1)]$$

$$g'(x) = (x-1)^2(x-2)(5x - 8)$$.

For $$g(x)$$ to be strictly decreasing, $$g'(x) < 0$$.

Critical points are $$x=1, x=2, x=8/5$$. Since $$(x-1)^2$$ is always positive, we look at $$(x-2)(5x-8) < 0$$.

This happens when $$x \in (8/5, 2)$$. So, $$\mathbf{b = 1.6}$$ and $$\mathbf{c = 2}$$.

$$100(4 + 1.6 - 2) = 100(3.6) = 360$$.

Correct Option: B (360)

We begin by considering a rectangle whose vertices are $$(0,y_1),\,(x,y_1),\,(x,y_2),\,(0,y_2)$$ subject to $$x\le y_1$$ and $$y_2\le 9-\frac{11}{3}x^2\,. $$

To maximize the area, it is natural to take the lower bound at $$y_1=x$$ and the upper bound at $$y_2=9-\frac{11}{3}x^2\,. $$

Area = $$x\bigl(9 - \frac{11}{3}x^2 - x\bigr)$$

$$= 9x - \frac{11}{3}x^3 - x^2$$

Next, we set the derivative of the area with respect to $$x$$ equal to zero:

$$\frac{dA}{dx} = 9 - 11x^2 - 2x = 0$$

$$11x^2 + 2x - 9 = 0$$

$$x = \frac{-2 \pm \sqrt{4+396}}{22} = \frac{-2 \pm 20}{22}$$

$$x = \frac{18}{22} = \frac{9}{11}$$ (taking the positive root)

Substituting $$x=\frac{9}{11}$$ back into the area formula gives

$$A = 9 \cdot \frac{9}{11} - \frac{11}{3}\left(\frac{9}{11}\right)^3 - \left(\frac{9}{11}\right)^2$$

The correct answer is Option 4: $$\frac{567}{121}$$.

Write the two inequalities separately.
  • From $$0 \le 9x \le y^{2}$$ we get $$x \ge 0$$ and $$y^{2} \ge 9x$$.
  • From $$y \ge 3x-6$$ we have the straight line $$y = 3x-6$$ (the region is above this line).

The parabola $$y^{2}=9x$$ can be written as two branches:
  Upper branch  $$y = 3\sqrt{x}$$,
  Lower branch  $$y = -3\sqrt{x}$$.

The condition $$y^{2}\ge 9x$$ means the point must lie outside the parabola, that is,
  either $$y \ge 3\sqrt{x}$$ or $$y \le -3\sqrt{x}$$.

The extra condition $$y \ge 3x-6$$ cuts away much of this unbounded region.
Let us find where the lower branch $$y=-3\sqrt{x}$$ meets the line $$y=3x-6$$:

Set them equal:
$$-3\sqrt{x}=3x-6$$   $$\Longrightarrow$$    divide by 3: $$-\sqrt{x}=x-2$$.
Put $$t=\sqrt{x}\; (t\ge 0)$$,
$$-t = t^{2}-2 \quad\Longrightarrow\quad t^{2}+t-2=0$$.
Solving, $$t=\dfrac{-1+\sqrt{1+8}}{2}=1 \;(\text{positive root})$$,
so $$\sqrt{x}=1 \Longrightarrow x=1$$ and the common point is $$(1,-3).$$

Next, compare the two curves for $$0\le x\le 1$$.
Take any $$x$$ in this interval (say $$x=0.25$$):
  • $$y=-3\sqrt{0.25}=-3\times 0.5=-1.5,$$
  • $$y=3(0.25)-6=-5.25.$$
Thus $$-3\sqrt{x} > 3x-6$$ in $$0\le x\le 1$$, so for these $$x$$ the region that satisfies
$$y\le -3\sqrt{x}\quad\text{and}\quad y\ge 3x-6$$ lies between the two curves.

For $$x\gt 1$$ the inequality reverses, giving no overlap between $$y\le -3\sqrt{x}$$ and $$y\ge 3x-6$$.
Hence the only bounded part of the region is the strip
$$0\le x\le 1,\quad 3x-6\le y\le -3\sqrt{x}.$$

Now compute its area $$A$$ by integrating with respect to $$x$$:

$$ \begin{aligned} A &= \int_{0}^{1}\Bigl[\;y_{\text{top}}-y_{\text{bottom}}\Bigr]\,dx \\ &= \int_{0}^{1}\Bigl[\,-3\sqrt{x}\;-\;(3x-6)\Bigr]\,dx \\ &= \int_{0}^{1}\!\bigl(-3x^{1/2}-3x+6\bigr)\,dx. \end{aligned} $$

Integrate term by term:

$$ \begin{aligned} A &= \left[-2x^{3/2}-\dfrac{3}{2}x^{2}+6x\right]_{0}^{1} \\ &= \Bigl(-2(1)^{3/2}-\dfrac{3}{2}(1)^{2}+6(1)\Bigr) \;-\;\Bigl(0+0+0\Bigr) \\ &= \bigl(-2 -1.5 +6\bigr)=2.5=\dfrac{5}{2}. \end{aligned} $$

Finally, the question asks for $$6A$$:

$$6A = 6 \times \dfrac{5}{2} = 15.$$

Therefore, $$6A = 15.$$

Given the function $$ f: (0, \infty) \to \mathbb{R} $$ is twice differentiable, and for some $$ a \neq 0 $$, the equation $$ \int_{0}^{1} f(\lambda x) d\lambda = a f(x) $$ holds, with $$ f(1) = 1 $$ and $$ f(16) = \frac{1}{8} $$. We need to find $$ 16 - f'\left( \frac{1}{16} \right) $$.

First, simplify the integral equation. Substitute $$ u = \lambda x $$, so $$ du = x d\lambda $$ and $$ d\lambda = \frac{du}{x} $$. When $$ \lambda = 0 $$, $$ u = 0 $$; when $$ \lambda = 1 $$, $$ u = x $$. Thus,
$$ \int_{0}^{1} f(\lambda x) d\lambda = \int_{0}^{x} f(u) \cdot \frac{du}{x} = \frac{1}{x} \int_{0}^{x} f(u) du $$
So the equation becomes:
$$ \frac{1}{x} \int_{0}^{x} f(u) du = a f(x) $$
Multiply both sides by $$ x $$:
$$ \int_{0}^{x} f(u) du = a x f(x) \quad \text{(1)} $$

Differentiate both sides of equation (1) with respect to $$ x $$. The left side, by the Fundamental Theorem of Calculus, is $$ f(x) $$. The right side, using the product rule, is $$ a \left[ f(x) + x f'(x) \right] $$. So,
$$ f(x) = a f(x) + a x f'(x) $$
Rearrange terms:
$$ f(x) - a f(x) = a x f'(x) $$
$$ f(x)(1 - a) = a x f'(x) $$
Thus,
$$ f'(x) = \frac{1 - a}{a} \cdot \frac{f(x)}{x} \quad \text{(2)} $$

Equation (2) is separable. Write it as:
$$ \frac{f'(x)}{f(x)} = \frac{1 - a}{a} \cdot \frac{1}{x} $$
Integrate both sides with respect to $$ x $$:
$$ \int \frac{f'(x)}{f(x)} dx = \frac{1 - a}{a} \int \frac{1}{x} dx $$
$$ \ln |f(x)| = \frac{1 - a}{a} \ln |x| + C $$
Since $$ x > 0 $$ and $$ f(x) $$ is defined for positive $$ x $$, drop the absolute values:
$$ \ln f(x) = \frac{1 - a}{a} \ln x + C $$
Exponentiate both sides:
$$ f(x) = e^C \cdot x^{\frac{1 - a}{a}} $$
Let $$ k = e^C $$, so
$$ f(x) = k x^{\frac{1 - a}{a}} \quad \text{(3)} $$

Use the given conditions. First, $$ f(1) = 1 $$:
$$ k \cdot (1)^{\frac{1 - a}{a}} = k \cdot 1 = k = 1 $$
Thus, $$ k = 1 $$, and
$$ f(x) = x^{\frac{1 - a}{a}} \quad \text{(4)} $$

Second, $$ f(16) = \frac{1}{8} $$:
$$ 16^{\frac{1 - a}{a}} = \frac{1}{8} $$
Express in terms of base 2: $$ 16 = 2^4 $$ and $$ \frac{1}{8} = 2^{-3} $$, so
$$ \left(2^4\right)^{\frac{1 - a}{a}} = 2^{-3} $$
$$ 2^{4 \cdot \frac{1 - a}{a}} = 2^{-3} $$
Equate exponents:
$$ 4 \cdot \frac{1 - a}{a} = -3 \quad \text{(5)} $$
Solve for $$ a $$:
$$ 4(1 - a) = -3a $$
$$ 4 - 4a = -3a $$
$$ 4 = 4a - 3a $$
$$ 4 = a $$
So $$ a = 4 $$.

Substitute $$ a = 4 $$ into equation (4):
$$ f(x) = x^{\frac{1 - 4}{4}} = x^{-\frac{3}{4}} $$

Now find the derivative:
$$ f(x) = x^{-\frac{3}{4}} $$
$$ f'(x) = -\frac{3}{4} x^{-\frac{3}{4} - 1} = -\frac{3}{4} x^{-\frac{7}{4}} $$

Evaluate at $$ x = \frac{1}{16} $$:
$$ f'\left( \frac{1}{16} \right) = -\frac{3}{4} \left( \frac{1}{16} \right)^{-\frac{7}{4}} $$
Note that $$ \left( \frac{1}{16} \right)^{-\frac{7}{4}} = 16^{\frac{7}{4}} $$, and $$ 16 = 2^4 $$, so
$$ 16^{\frac{7}{4}} = (2^4)^{\frac{7}{4}} = 2^{4 \cdot \frac{7}{4}} = 2^7 = 128 $$
Thus,
$$ f'\left( \frac{1}{16} \right) = -\frac{3}{4} \cdot 128 = -\frac{384}{4} = -96 $$

Finally, compute:
$$ 16 - f'\left( \frac{1}{16} \right) = 16 - (-96) = 16 + 96 = 112 $$

The answer is 112.

For $$x \neq 0$$ the function is $$f(x)=x^{3}+ax^{2}+b\,\log_e|x|+1$$.

Critical points satisfy $$f'(x)=0$$.
Differentiate: $$f'(x)=3x^{2}+2ax+\dfrac{b}{x}$$.

The points $$x=-1$$ and $$x=2$$ are given to be critical, so

At $$x=-1$$: $$3(-1)^{2}+2a(-1)+\dfrac{b}{-1}=0 \;\;\Longrightarrow\;\; 3-2a-b=0$$ $$\Rightarrow\; 2a+b=3 \;-(1)$$

At $$x=2$$: $$3(2)^{2}+2a(2)+\dfrac{b}{2}=0 \;\;\Longrightarrow\;\; 12+4a+\dfrac{b}{2}=0$$ Multiply by $$2$$: $$24+8a+b=0 \;-(2)$$

From $$(2)$$ we get $$b=-24-8a$$.
Put this in $$(1)$$:

$$2a+(-24-8a)=3 \;\;\Longrightarrow\;\; -6a=27 \;\;\Longrightarrow\;\; a=-\dfrac{9}{2}$$

Hence $$b=-24-8\left(-\dfrac{9}{2}\right)=12$$.

The required interval is $$\left[-2,\,-\dfrac{1}{2}\right]$$. Inside it, possible locations of absolute extrema are:

• the critical point $$x=-1$$ (it lies inside the interval)
• the endpoints $$x=-2$$ and $$x=-\dfrac{1}{2}$$.

Substitute $$a=-\dfrac{9}{2},\; b=12$$ in $$f(x)$$:

$$f(x)=x^{3}-\dfrac{9}{2}x^{2}+12\log_e|x|+1$$.

Case 1:

$$x=-1$$:
$$f(-1)=(-1)^{3}-\dfrac{9}{2}(1)+12\log_e 1+1$$ $$f(-1)=-1-\dfrac{9}{2}+0+1=-\dfrac{9}{2}=-4.5$$

Case 2:

$$x=-2$$:
$$f(-2)=(-8)-\dfrac{9}{2}(4)+12\log_e 2+1$$ $$f(-2)=-8-18+12\log_e 2+1=-25+12\log_e 2$$

Case 3:

$$x=-\dfrac{1}{2}$$:
$$f\!\left(-\dfrac{1}{2}\right)=\left(-\dfrac{1}{8}\right)-\dfrac{9}{2}\!\left(\dfrac{1}{4}\right)+12\log_e\!\left(\dfrac{1}{2}\right)+1$$ $$= -\dfrac{1}{8}-\dfrac{9}{8}-12\log_e 2+1$$ $$= -\dfrac{10}{8}-12\log_e 2+1=-\dfrac{5}{4}-12\log_e 2+1$$ $$= -\dfrac{1}{4}-12\log_e 2$$ (numerically about $$-8.57$$)

Compare the three values:

$$f(-2)=-25+12\log_e 2 \approx -16.68$$ (smallest)
$$f\!\left(-\dfrac{1}{2}\right)\approx -8.57$$
$$f(-1)=-4.5$$ (largest)

Therefore
Absolute minimum $$m=f(-2)=-25+12\log_e 2$$,
Absolute maximum $$M=f(-1)=-\dfrac{9}{2}$$.

Compute $$|M+m|$$:

$$M+m=-\dfrac{9}{2}+\bigl(-25+12\log_e 2\bigr)=-\dfrac{59}{2}+12\log_e 2$$ Since this value is negative, $$|M+m|=\dfrac{59}{2}-12\log_e 2$$.

Numerically: $$\dfrac{59}{2}=29.5,\; 12\log_e 2\approx 8.3178$$ $$|M+m|\approx 29.5-8.3178\approx 21.18$$.

This matches Option A (21.1). Hence

$$|M+m| \approx 21.1$$ (Option A).

By the Leibniz rule (chain rule with integral) we have $$f'(x) = \frac{(x^2)^2 - 8(x^2) + 15}{e^{x^2}}\cdot 2x = \frac{2x\bigl(x^4 - 8x^2 + 15\bigr)}{e^{x^2}}\,. $$ Factoring the quartic by setting $$u = x^2$$ gives $$x^4 - 8x^2 + 15 = u^2 - 8u + 15 = (u - 3)(u - 5) = (x^2 - 3)(x^2 - 5)\,. $$ This yields $$f'(x) = \frac{2x\,(x^2 - 3)\,(x^2 - 5)}{e^{x^2}}\,. $$

Since $$e^{x^2}\gt 0$$ always, the critical points occur when the numerator is zero, namely when $$2x=0$$ or $$x^2-3=0$$ or $$x^2-5=0\,. $$ Hence the critical points are $$x=0,\quad x=\pm\sqrt{3},\quad x=\pm\sqrt{5}\,, $$ giving five critical values in all.

To determine the nature of each critical point, we examine the sign of $$f'(x)=\frac{2x\,(x^2-3)\,(x^2-5)}{e^{x^2}}\,. $$ For $$x\lt -\sqrt{5}$$ all three factors $$2x$$, $$(x^2-3)$$ and $$(x^2-5)$$ have signs $$-$$, $$+$$, $$+$$ respectively, so $$f'(x)\lt 0\,. $$ In the interval $$-\sqrt{5} \lt x \lt -\sqrt{3}$$ the factors are $$(-){\cdot}(+){\cdot}(-)$$ so $$f'(x) \gt 0\,. $$ For $$-\sqrt{3} \lt x \lt 0$$ the signs are $$(-){\cdot}(-){\cdot}(-)$$ giving $$f'(x) \lt 0\,, $$ and for $$0 \lt x \lt \sqrt{3}$$ they are $$(+){\cdot}(-){\cdot}(-)$$ giving $$f'(x) \gt 0\,. $$ In $$\sqrt{3} \lt x \lt \sqrt{5}$$ the signs are $$(+){\cdot}(+){\cdot}(-)$$ so $$f'(x) \lt 0\,, $$ while for $$x \gt \sqrt{5}$$ all three factors are positive and $$f'(x)\gt 0\,. $$

Since $$f'(x)$$ changes from negative to positive at $$x=-\sqrt{5},\;0,\;\sqrt{5}$$ these are local minima, and since it changes from positive to negative at $$x=-\sqrt{3},\;\sqrt{3}$$ these are local maxima. Thus there are 2 local maxima and 3 local minima. The answer is Option A: 2 and 3.

We are given $$f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$$ where $$a \gt 0$$.

Finding critical points: $$f'(x) = 6x^2 - 18ax + 12a^2 = 6(x^2 - 3ax + 2a^2) = 6(x - a)(x - 2a)$$.

So the critical points are $$x = a$$ and $$x = 2a$$.

Since the coefficient of $$x^3$$ is positive, $$f$$ has a local maximum at $$x = a$$ and a local minimum at $$x = 2a$$. Therefore $$p = a$$ and $$q = 2a$$.

Given $$p^2 = q$$: $$a^2 = 2a$$, so $$a(a - 2) = 0$$. Since $$a \gt 0$$, we get $$a = 2$$.

Now $$f(x) = 2x^3 - 18x^2 + 48x + 1$$.

$$f(3) = 2(27) - 18(9) + 48(3) + 1 = 54 - 162 + 144 + 1 = 37$$.

Hence, the correct answer is Option D.

One vertex is at the origin $$(0,0)$$, and the other two are at $$(x, y)$$ and $$(-x, y)$$ on the curve $$y = -2x^2 + 54$$ where $$y > 0$$.

The base of the triangle = $$2x$$, and the height = $$y$$.

Area = $$\frac{1}{2} \times 2x \times y = xy = x(-2x^2 + 54) = -2x^3 + 54x$$.

To maximize, take the derivative and set it to zero:

$$ \frac{dA}{dx} = -6x^2 + 54 = 0 \Rightarrow x^2 = 9 \Rightarrow x = 3 $$

(taking positive value since $$x > 0$$)

$$ y = -2(9) + 54 = -18 + 54 = 36 $$

Maximum area = $$3 \times 36 = 108$$.

Verify it's a maximum: $$\frac{d^2A}{dx^2} = -12x = -36 < 0$$. Confirmed maximum.

The answer is Option (4): $$\boxed{108}$$.

Given ( $$f(x)=\cos x-x+1$$) on$$([0,\pi])$$

(S1): (f(x)=0) has only one solution in ([0,\pi])

Check endpoints:
$$f(0)=1-0+1=2>0,\quad f(\pi)=\cos\pi-\pi+1=-1-\pi+1=-\pi<0$$

Since (f) is continuous, there is at least one root.

Now derivative:
$$f'(x)=-\sin x-1$$

$$Since(\sin x\ge0)on([0,\pi]),$$
$$f'(x)=-(\sin x+1)<0$$

So (f) is strictly decreasing, hence it can cross zero only once.

(S1) is true 

(S2): decreasing in ([0,\pi/2]) and increasing in ([\pi/2,\pi])

But we already have:
$$f'(x)<0\quad\text{for all }x\in[0,\pi]$$

So (f) is decreasing everywhere — never increasing.

 (S2) is false

Only (S1) is correct

We are given that $$f(x)$$ is maximized at $$x = 1/e$$. This implies that for any $$x \neq 1/e$$, $$f(1/e) > f(x)$$.

Let's look at a simpler related function $$g(x) = x^{1/x}$$. It is well known (via $$g'(x)$$) that $$x^{1/x}$$ is maximized at $$x = e$$.

Step 2: Compare the values.

Since $$e < \pi$$, and the function $$g(x) = x^{1/x}$$ is strictly decreasing for $$x > e$$:

$$e^{1/e} > \pi^{1/\pi}$$

Raise both sides to the power of $$e\pi$$:

$$(e^{1/e})^{e\pi} > (\pi^{1/\pi})^{e\pi} \implies e^\pi > \pi^e$$

Correct Option: B ($$e^\pi > \pi^e$$)

We are told that $$f''(x) \gt 0$$ for every $$x \in (0,3)$$.
A positive second derivative means $$f(x)$$ is strictly convex and its first derivative $$f'(x)$$ is strictly increasing on $$(0,3)$$.

Define $$g(x)=3\,f\!\left(\tfrac{x}{3}\right)+f(3-x)\,.$$

Step 1 : Differentiate $$g(x)$$
Use the chain rule term-by-term:

$$\frac{d}{dx}\Bigl[3\,f\!\left(\tfrac{x}{3}\right)\Bigr] = 3\;f'\!\left(\tfrac{x}{3}\right)\cdot\frac{1}{3} = f'\!\left(\tfrac{x}{3}\right)$$

$$\frac{d}{dx}\Bigl[f(3-x)\Bigr] = f'(3-x)\cdot(-1) = -\,f'(3-x)$$

Therefore

$$g'(x)=f'\!\left(\tfrac{x}{3}\right)-f'(3-x)\,.$$

Step 2 : Locate the point $$\alpha$$ where $$g(x)$$ switches from decreasing to increasing
The description “$$g$$ is decreasing on $$(0,\alpha)$$ and increasing on $$(\alpha,3)$$” means $$g'(x)\lt 0$$ for $$0\lt x\lt\alpha$$, $$g'(\alpha)=0$$, and $$g'(x)\gt 0$$ for $$\alpha\lt x\lt3$$.
Hence $$g'(\alpha)=0$$.

Set $$g'(\alpha)=0$$:

$$f'\!\left(\tfrac{\alpha}{3}\right)-f'(3-\alpha)=0 \;\Longrightarrow\; f'\!\left(\tfrac{\alpha}{3}\right)=f'(3-\alpha)\,.$$

Step 3 : Use the strict monotonicity of $$f'(x)$$
Because $$f'(x)$$ is strictly increasing on $$(0,3)$$, it is a one-to-one (injective) function there.
If two inputs give the same output, the inputs themselves must be equal:

$$\tfrac{\alpha}{3}=3-\alpha\,.$$

Step 4 : Solve for $$\alpha$$

Multiply both sides by $$3$$: $$\alpha = 9-3\alpha$$.
Bring like terms together: $$4\alpha = 9$$.
Hence $$\alpha = \dfrac{9}{4}=2.25$$, which indeed lies inside $$(0,3)$$.

Step 5 : Compute $$8\alpha$$

$$8\alpha = 8 \times \dfrac{9}{4} = 18.$$

Therefore $$8\alpha = 18$$, which corresponds to Option C.

We need to determine the local maxima and minima of $$f(x) = 2x + 3x^{2/3}$$ for $$x \in \mathbb{R}$$.

$$f'(x) = 2 + 3 \cdot \frac{2}{3}\, x^{-1/3} = 2 + \frac{2}{x^{1/3}}$$

This can be written as:

$$f'(x) = \frac{2x^{1/3} + 2}{x^{1/3}} = \frac{2(x^{1/3} + 1)}{x^{1/3}}$$

Critical points occur where $$f'(x) = 0$$ or $$f'(x)$$ is undefined.

Case 1: $$f'(x) = 0$$: The numerator $$2(x^{1/3} + 1) = 0 \implies x^{1/3} = -1 \implies x = -1$$.

Case 2: $$f'(x)$$ undefined: The denominator $$x^{1/3} = 0 \implies x = 0$$.

So the critical points are $$x = -1$$ and $$x = 0$$.

At $$x = -1$$: $$f'(x)$$ changes from $$+ve$$ to $$-ve$$ → local maximum.

$$f(-1) = 2(-1) + 3(-1)^{2/3} = -2 + 3(1) = 1$$

At $$x = 0$$: $$f'(x)$$ changes from $$-ve$$ to $$+ve$$ → local minimum.

$$f(0) = 0$$

Conclusion: The function has exactly one point of local maxima (at $$x = -1$$) and exactly one point of local minima (at $$x = 0$$).

The answer is Option C: exactly one local maxima and exactly one local minima.

 $$f(x) = (x+3)^2(x-2)^3$$ on $$[-4, 4]$$.

$$f'(x) = 2(x+3)(x-2)^3 + 3(x+3)^2(x-2)^2 = (x+3)(x-2)^2[2(x-2)+3(x+3)] = (x+3)(x-2)^2(5x+5) = 5(x+3)(x-2)^2(x+1)$$

Critical points: $$x = -3, -1, 2$$.

$$f(-4) = (-1)^2(-6)^3 = -216$$. $$f(-3) = 0$$.

$$f(-1) = (2)^2(-3)^3 = 4(-27) = -108$$. $$f(2) = 0$$. $$f(4) = (7)^2(2)^3 = 49(8) = 392$$.

Maximum $$M = 392$$ at $$x = 4$$. Minimum $$m = -216$$ at $$x = -4$$.

$$M - m = 392 - (-216) = 608$$.

Consider the function $$f(x) = \frac{x}{x^2 - 6x - 16}$$ whose domain is $$\mathbb{R} - \{-2, 8\}$$. To determine its monotonicity, we compute its derivative using the quotient rule:

$$ f'(x) = \frac{(x^2 - 6x - 16)\cdot 1 - x\cdot (2x - 6)}{(x^2 - 6x - 16)^2}. $$

The numerator simplifies as follows:

$$ x^2 - 6x - 16 - 2x^2 + 6x = -x^2 - 16 = -(x^2 + 16). $$

Therefore,

$$ f'(x) = \frac{-(x^2 + 16)}{(x^2 - 6x - 16)^2}. $$

Since $$x^2 + 16 > 0$$ for all real $$x$$, the numerator is always negative, and the denominator $$(x^2 - 6x - 16)^2$$ is always positive (being a perfect square and nonzero in the domain). Hence $$f'(x) < 0$$ on every interval of its domain.

It follows that $$f(x)$$ is decreasing on $$(-\infty, -2)\cup(-2, 8)\cup(8, \infty)$$.

The correct answer is Option (2).

Find $$\alpha^2 + \beta^2$$ where $$(\alpha, \beta)$$ is the set of positive $$\lambda$$ values for which the local minimum of $$(1 + x(\lambda^2 - x^2))$$ satisfies $$\frac{x^2+x+2}{x^2+5x+6} < 0$$.

$$f'(x) = \lambda^2 - 3x^2 = 0 \Rightarrow x = \pm \frac{\lambda}{\sqrt{3}}$$

$$f''(x) = -6x$$. At $$x = \lambda/\sqrt{3}$$: $$f'' = -6\lambda/\sqrt{3} < 0$$ (local max). At $$x = -\lambda/\sqrt{3}$$: $$f'' = 6\lambda/\sqrt{3} > 0$$ (local min).

The local minimum is at $$x_0 = -\frac{\lambda}{\sqrt{3}}$$ (negative value since $$\lambda > 0$$).

$$\frac{x_0^2 + x_0 + 2}{x_0^2 + 5x_0 + 6} < 0$$

Numerator: $$x_0^2 + x_0 + 2$$. Discriminant = $$1 - 8 = -7 < 0$$. Since leading coefficient > 0, numerator is always positive.

Denominator: $$x_0^2 + 5x_0 + 6 = (x_0+2)(x_0+3) < 0$$

This requires $$-3 < x_0 < -2$$.

$$-3 < -\frac{\lambda}{\sqrt{3}} < -2$$

$$2 < \frac{\lambda}{\sqrt{3}} < 3$$

$$2\sqrt{3} < \lambda < 3\sqrt{3}$$

So $$\alpha = 2\sqrt{3}$$, $$\beta = 3\sqrt{3}$$.

$$\alpha^2 + \beta^2 = 12 + 27 = 39$$

The correct answer is $$\boxed{39}$$.

$$f(x)=(p^2-6p+8)(\sin^22x-\cos^22x)+2(2-p)x+7$$

$$\sin^22x-\cos^22x=-\cos4x$$
$$\Rightarrow f'(x)=4(p^2-6p+8)\sin4x+2(2-p)$$
For no critical point:
$$4(p^2-6p+8)\sin4x+2(2-p)\ne0$$

$$\Rightarrow\left|\frac{2(p-2)}{4(p^2-6p+8)}\right|>1$$
$$\Rightarrow|p-2|>2|p^2-6p+8|$$

$$(p^2-6p+8)=(p-2)(p-4)$$
$$\Rightarrow|p-2|>2|p-2||p-4|$$

$$For(p\ne2):$$
$$1>2|p-4|\Rightarrow|p-4|<\frac{1}{2}$$

$$\Rightarrow p\in\left(\frac{7}{2},\frac{9}{2}\right)$$

16ab=16$$\cdot$$ $$\frac{7}{2}\cdot$$ $$\frac{9}{2}$$=252

Integrating $$f''(x) = g''(x) + 6x$$ gives $$f'(x) = g'(x) + 3x^2 + C_1$$. From $$f'(1) = 9$$ and $$4g'(1) - 3 = 9 \Rightarrow g'(1) = 3$$: $$9 = 3 + 3 + C_1$$, so $$C_1 = 3$$. Thus $$f'(x) - g'(x) = 3x^2 + 3$$.

Integrating again: $$f(x) - g(x) = x^3 + 3x + C_2$$. Using $$f(2) = 3$$ and $$g(2) = 12$$: $$3 - 12 = 8 + 6 + C_2$$, giving $$C_2 = -23$$. Hence $$f(x) - g(x) = x^3 + 3x - 23$$.

Option (B) states: if $$-1 < x < 2$$, then $$|f(x) - g(x)| < 8$$. Since $$h(x) = x^3 + 3x - 23$$ is strictly increasing ($$h'(x) = 3x^2 + 3 > 0$$), its values on $$(-1, 2)$$ range from $$h(-1) = -27$$ to $$h(2) = -9$$. So $$|f(x) - g(x)|$$ ranges from $$9$$ to $$27$$, which is always greater than $$8$$. Option (B) is NOT true.

The answer is $$\boxed{\text{Option (B)}}$$.

A square piece of tin of side 30 cm. Cut squares of side $$x$$ from each corner.

After folding: length = width = $$30 - 2x$$, height = $$x$$.

Volume: $$V = x(30-2x)^2$$

$$\frac{dV}{dx} = (30-2x)^2 + x \cdot 2(30-2x)(-2) = (30-2x)[(30-2x) - 4x] = (30-2x)(30-6x)$$

Setting $$\frac{dV}{dx} = 0$$: $$x = 15$$ (rejected, gives V=0) or $$x = 5$$.

At $$x = 5$$: length = width = 20 cm, height = 5 cm.

Surface area (without top) = base + 4 sides:

$$= 20 \times 20 + 4 \times (20 \times 5) = 400 + 400 = 800 \text{ cm}^2$$

The correct answer is Option 1: 800.

$$f(x) = 2x^3 + (2p - 7)x^2 + 3(2p - 9)x - 6$$

$$f'(x) = 6x^2 + 2(2p - 7)x + 3(2p - 9)$$

A local maxima for some value of $$x < 0$$ means a negative root

A local minima for some value of $$x > 0$$ means a positive root

$$6x^2 + 2(2p - 7)x + 3(2p - 9) = 0$$

$$\frac{C}{A} < 0 \implies \frac{3(2p - 9)}{6} < 0$$ (For roots to be of opposite signs)

$$2p < 9 \implies p < \frac{9}{2}$$

$$p \in \left(-\infty, \frac{9}{2}\right)$$

$$f'(x) = 8x^3 - 36x + 8$$

Set $$f'(x) = 0$$: $$8x^3 - 36x + 8 = 0$$

$$2x^3 - 9x + 2 = 0$$

$$2x^3 - 9x + 2 = (x - 2)(2x^2 + 4x - 1) = 0$$

$$x = \frac{-4 \pm \sqrt{16 - 4(2)(-1)}}{2(2)} = \frac{-4 \pm \sqrt{24}}{4} = \frac{-4 \pm 2\sqrt{6}}{4} = -1 \pm \frac{\sqrt{6}}{2}$$

Our critical points are $$x = 2$$, $$x = -1 + \frac{\sqrt{6}}{2} \approx 0.22$$, and $$x = -1 - \frac{\sqrt{6}}{2} \approx -2.22$$.

Using the second derivative test, $$f''(x) = 24x^2 - 36$$:

$$f''(2) = 24(4) - 36 = 60 > 0$$ (Confirms local minima at $$x=2$$).

$$f''(-1 - \frac{\sqrt{6}}{2}) > 0$$ (This is another local minima).

$$f''(-1 + \frac{\sqrt{6}}{2}) = 24(-1 + \frac{\sqrt{6}}{2})^2 - 36$$. Since $$(-1 + \frac{\sqrt{6}}{2})^2 \approx 0.05$$, $$f'' < 0$$ (This is the local maxima).

$$f(-1 + \frac{\sqrt{6}}{2})$$ = $$12\sqrt{6} - \frac{33}{2}$$

We need to find $$a + 2b + 7$$ given that $$f(x)$$ and $$g(x)$$ have a common extreme point.

$$f'(x) = x^2 + ax + 2b$$

$$g'(x) = x^2 + 2bx + a$$

$$f'(c) = c^2 + ac + 2b = 0 \quad \text{...(i)}$$

$$g'(c) = c^2 + 2bc + a = 0 \quad \text{...(ii)}$$

$$(a - 2b)c + (2b - a) = 0$$

$$(a - 2b)(c - 1) = 0$$

Since $$a \neq 2b$$, we must have $$c = 1$$.

$$1 + a + 2b = 0 \implies a + 2b = -1$$

$$a + 2b + 7 = -1 + 7 = 6$$

The correct answer is Option D: $$6$$.

We need to find the maximum value of $$f(x) = \left(\frac{\sqrt{3e}}{2\sin x}\right)^{\sin^2 x}$$ for $$x \in \left(0, \frac{\pi}{2}\right)$$.

Let $$g(x) = \ln f(x) = \sin^2 x \cdot \ln\left(\frac{\sqrt{3e}}{2\sin x}\right)$$.

Let $$t = \sin x$$ where $$t \in (0, 1)$$. Then:

$$ g = t^2 \left[\frac{1}{2}\ln(3e) - \ln(2t)\right] = t^2 \left[\frac{1}{2}\ln 3 + \frac{1}{2} - \ln 2 - \ln t\right] $$

Differentiating with respect to $$t$$:

$$ \frac{dg}{dt} = 2t\left[\frac{1}{2}\ln(3e) - \ln(2t)\right] + t^2 \cdot \left(-\frac{1}{t}\right) $$

$$ = t\left[\ln\left(\frac{3e}{4t^2}\right) - 1\right] $$

Setting $$\frac{dg}{dt} = 0$$ (with $$t > 0$$):

$$ \ln\left(\frac{3e}{4t^2}\right) = 1 \implies \frac{3e}{4t^2} = e \implies t^2 = \frac{3}{4} \implies t = \frac{\sqrt{3}}{2} $$

This corresponds to $$\sin x = \frac{\sqrt{3}}{2}$$, i.e., $$x = \frac{\pi}{3}$$.

At $$t = \frac{\sqrt{3}}{2}$$:

$$ f = \left(\frac{\sqrt{3e}}{2 \cdot \frac{\sqrt{3}}{2}}\right)^{3/4} = \left(\frac{\sqrt{3e}}{\sqrt{3}}\right)^{3/4} = \left(\sqrt{e}\right)^{3/4} = e^{3/8} $$

So the maximum value of $$f(x)$$ is $$\frac{k}{e} = e^{3/8}$$, giving $$k = e^{11/8}$$.

$$\left(\frac{k}{e}\right)^8 = \left(e^{3/8}\right)^8 = e^3$$

$$\frac{k^8}{e^5} = \frac{\left(e^{11/8}\right)^8}{e^5} = \frac{e^{11}}{e^5} = e^6$$

$$k^8 = \left(e^{11/8}\right)^8 = e^{11}$$

Therefore: $$\left(\frac{k}{e}\right)^8 + \frac{k^8}{e^5} + k^8 = e^3 + e^6 + e^{11}$$

The correct answer is Option A: $$e^3 + e^6 + e^{11}$$.

Step 1: Find the slope of the given line

The equation of the given line is:

$$x + 90y + 2 = 0$$

Rewriting it in slope-intercept form ($$y = mx + c$$):

$$90y = -x - 2 \implies y = -\frac{1}{90}x - \frac{2}{90}$$

Thus, the slope of this line is $$m = -\frac{1}{90}$$.

Step 2: Formulate the condition for parallel normal lines

Since the normal lines to the curve are parallel to the given line, their slope must be equal to the slope of the line:

$$m_{\text{normal}} = -\frac{1}{90}$$

The relationship between the slope of the tangent ($$\frac{dy}{dx}$$) and the slope of the normal is given by:

$$m_{\text{normal}} = -\frac{1}{\frac{dy}{dx}} = -\frac{1}{90} \implies \frac{dy}{dx} = 90$$

Step 3: Differentiate the given curve equation

The equation of the curve is:

$$y = 54x^5 - 135x^4 - 70x^3 + 180x^2 + 210x$$

Differentiating $$y$$ with respect to $$x$$ gives:

$$\frac{dy}{dx} = 54(5x^4) - 135(4x^3) - 70(3x^2) + 180(2x) + 210$$

$$\frac{dy}{dx} = 270x^4 - 540x^3 - 210x^2 + 360x + 210$$

Step 4: Set the derivative equal to 90 and solve the polynomial equation

Substituting $$\frac{dy}{dx} = 90$$ into the derivative equation:

$$270x^4 - 540x^3 - 210x^2 + 360x + 210 = 90$$

Subtracting 90 from both sides:

$$270x^4 - 540x^3 - 210x^2 + 360x + 120 = 0$$

Dividing the entire equation by 30 to simplify:

$$9x^4 - 18x^3 - 7x^2 + 12x + 4 = 0$$

By inspection, we test integer roots:

- For $$x = 1$$: $$9(1)^4 - 18(1)^3 - 7(1)^2 + 12(1) + 4 = 9 - 18 - 7 + 12 + 4 = 0$$ (So, $$x = 1$$ is a root)

- For $$x = 2$$: $$9(2)^4 - 18(2)^3 - 7(2)^2 + 12(2) + 4 = 144 - 144 - 28 + 24 + 4 = 0$$ (So, $$x = 2$$ is a root)

Since $$x = 1$$ and $$x = 2$$ are roots, $$(x-1)(x-2) = x^2 - 3x + 2$$ is a factor. Dividing the polynomial by this factor yields the remaining quadratic equation:

$$9x^2 + 9x + 2 = 0$$

Factoring the quadratic equation:

$$(3x + 1)(3x + 2) = 0 \implies x = -\frac{1}{3} \quad \text{or} \quad x = -\frac{2}{3}$$

The four distinct real roots for the x-coordinates are $$x = 1, 2, -\frac{1}{3}, -\frac{2}{3}$$. Since all four values are real and distinct, they correspond to four unique points on the curve.

Conclusion:

The number of points on the curve at which the normal lines are parallel to the given line is 4.

Given $$f(x)=\left|x^{2}-5x+6\right|-3x+2$$ on the closed interval $$[-1,\,3]$$. Write the quadratic inside the modulus in factored form:

$$x^{2}-5x+6=(x-2)(x-3)$$

Its zeros are $$x=2$$ and $$x=3$$, which lie inside $$[-1,3]$$. Hence the sign of $$x^{2}-5x+6$$ is:

  • positive on $$[-1,\,2]$$, because a test point $$x=0$$ gives $$6\gt0$$.
  • negative on $$(2,\,3]$$, because a test point $$x=2.5$$ gives $$-0.25\lt0$$.

Therefore split $$f(x)$$ into two smooth pieces.

Case 1:

For $$x\in[-1,\,2]$$, $$|x^{2}-5x+6|=x^{2}-5x+6$$, so

$$f_1(x)=x^{2}-5x+6-3x+2=x^{2}-8x+8.$$ Find extrema of $$f_1(x)$$ on $$[-1,\,2]$$.

Derivative: $$f_1'(x)=2x-8.$$ Set $$f_1'(x)=0\ \Rightarrow\ 2x-8=0\ \Rightarrow\ x=4,$$ which lies outside $$[-1,2]$$. Thus only the endpoints need checking.

$$f_1(-1)=(-1)^{2}-8(-1)+8=1+8+8=17$$
$$f_1(2)=2^{2}-8\cdot2+8=4-16+8=-4$$

On $$[-1,2]$$: absolute maximum $$=17$$, absolute minimum $$=-4$$.

Case 2:

For $$x\in[2,\,3]$$, $$|x^{2}-5x+6|=-(x^{2}-5x+6)=-x^{2}+5x-6,$$ so

$$f_2(x)=-x^{2}+5x-6-3x+2=-x^{2}+2x-4.$$ Find extrema of $$f_2(x)$$ on $$[2,\,3]$$.

Derivative: $$f_2'(x)=-2x+2.$$ Set $$f_2'(x)=0\ \Rightarrow\ -2x+2=0\ \Rightarrow\ x=1,$$ which lies outside $$[2,3]$$. Again, check only the endpoints.

$$f_2(2)=-(2)^{2}+2\cdot2-4=-4+4-4=-4$$
$$f_2(3)=-(3)^{2}+2\cdot3-4=-9+6-4=-7$$

On $$[2,3]$$: absolute maximum $$=-4$$, absolute minimum $$=-7$$.

Combining both cases, on the whole interval $$[-1,3]$$ we have absolute maximum value $$=17$$ (at $$x=-1$$) and absolute minimum value $$=-7$$ (at $$x=3$$).

Required sum = $$17+(-7)=10$$.

Hence the correct option is Option A $$(10)$$.

A wire of length 20 m is cut into two pieces. Piece of length $$\ell_1$$ forms a square, and piece of length $$\ell_2$$ forms a circle, where $$\ell_1 + \ell_2 = 20$$.

Area of the square: $$A_1 = \left(\frac{\ell_1}{4}\right)^2 = \frac{\ell_1^2}{16}$$

Area of the circle: Circumference = $$\ell_2 = 2\pi r$$, so $$r = \frac{\ell_2}{2\pi}$$ and $$A_2 = \pi r^2 = \frac{\ell_2^2}{4\pi}$$

Minimize: $$f = 2A_1 + 3A_2 = \frac{\ell_1^2}{8} + \frac{3\ell_2^2}{4\pi}$$

Substituting $$\ell_2 = 20 - \ell_1$$ and differentiating:

$$ f'(\ell_1) = \frac{\ell_1}{4} - \frac{3(20 - \ell_1)}{2\pi} = 0 $$

$$ \frac{\pi\ell_1}{4} = \frac{3(20 - \ell_1)}{2} $$

$$ \frac{\pi\ell_1}{2} = 3(20 - \ell_1) = 60 - 3\ell_1 $$

$$ \pi\ell_1 + 6\ell_1 = 120 $$

$$ \ell_1 = \frac{120}{\pi + 6}, \quad \ell_2 = 20 - \frac{120}{\pi + 6} = \frac{20\pi}{\pi + 6} $$

The ratio:

$$ \frac{\pi\ell_1}{\ell_2} = \frac{\pi \times \frac{120}{\pi+6}}{\frac{20\pi}{\pi+6}} = \frac{120}{20} = 6 $$

Therefore, $$\pi\ell_1 : \ell_2 = 6 : 1$$.

To find $$A$$, we integrate $$f(x) = \min \{x^2 + \frac{3}{4}, 1 + [x]\}$$ within the region bounded by $$x+y=2$$ (i.e., $$y = 2-x$$) and the axes.

1. Breakdown of $$f(x)$$ for $$x \in [0, 2]$$

• $$x \in [0, \frac{1}{2}]$$: $$x^2 + \frac{3}{4} \le 1$$, so $$f(x) = x^2 + \frac{3}{4}$$
• $$x \in [\frac{1}{2}, 1]$$: $$x^2 + \frac{3}{4} > 1$$, so $$f(x) = 1$$
• $$x \in [1, 2]$$: Here $$1 + [x] = 2$$. Since the line $$y = 2-x$$ is always $$\le 2$$ in this range, the boundary of the enclosed area is $$y = 2-x$$.

2. Integration for Area $$A$$

The area $$A$$ is the integral of the lower boundary:

$$A = \int_{0}^{1/2} (x^2 + \frac{3}{4}) dx + \int_{1/2}^{1} 1 dx + \int_{1}^{2} (2-x) dx$$

Calculations:

1. $$\int_{0}^{1/2} (x^2 + \frac{3}{4}) dx = [\frac{x^3}{3} + \frac{3x}{4}]_0^{1/2} = \frac{1}{24} + \frac{3}{8} = \frac{10}{24} = \frac{5}{12}$$
2. $$\int_{1/2}^{1} 1 dx = 1 - \frac{1}{2} = \frac{1}{2}$$
3. $$\int_{1}^{2} (2-x) dx = [2x - \frac{x^2}{2}]_1^2 = (4 - 2) - (2 - \frac{1}{2}) = 2 - 1.5 = \frac{1}{2}$$

3. Final Value

$$A = \frac{5}{12} + \frac{1}{2} + \frac{1}{2} = \frac{5}{12} + 1 = \frac{17}{12}$$

Therefore:

$$12A = 12 \times \frac{17}{12} = \mathbf{17}$$

Quadratic $$f(x) = ax^2+bx+c$$ passes through (-1,0) and touches y=x at (1,1).

f(-1)=0: a-b+c=0. f(1)=1: a+b+c=1. f'(1)=1: 2a+b=1.

From first two: 2a+2c=1. From third: b=1-2a. From first: a-(1-2a)+c=0, 3a+c=1.

Also 2a+2c=1, so c=(1-2a)/2. Substituting: 3a+(1-2a)/2=1, (6a+1-2a)/2=1, 4a=1, a=1/4.

b=1/2, c=1/4. So f(x)=(x²+2x+1)/4=(x+1)²/4.

f(α)=α+1: (α+1)²/4=α+1, α+1=4 (since α+1≠0 in first quadrant), α=3.

f'(3)=(3+1)/2=2. Normal slope=-1/2. Normal at (3,4): y-4=-1/2(x-3).

x-intercept: y=0: -4=-1/2(x-3), x-3=8, x=11.

The answer is 11.

We need to find the number of points where $$y = x^5 - 20x^3 + 50x + 2$$ crosses the x-axis.

Let $$f(x) = x^5 - 20x^3 + 50x + 2$$.

Find critical points.

$$f'(x) = 5x^4 - 60x^2 + 50 = 5(x^4 - 12x^2 + 10) = 0$$

Treating as a quadratic in $$x^2$$:

$$x^2 = \frac{12 \pm \sqrt{144 - 40}}{2} = 6 \pm \sqrt{26}$$

$$\sqrt{26} \approx 5.099$$

$$x^2 \approx 11.099 \text{ or } 0.901$$

Critical points: $$x \approx \pm 3.331$$ and $$x \approx \pm 0.949$$.

Evaluate $$f$$ at critical points.

$$f(-3.331) \approx 164.55 > 0$$

$$f(-0.949) \approx -29.13 < 0$$

$$f(0.949) \approx 33.13 > 0$$

$$f(3.331) \approx -160.55 < 0$$

Count sign changes using the Intermediate Value Theorem.

Since $$f$$ is continuous and changes sign between consecutive critical values and at the extremes:

1. $$f(x) \to -\infty$$ as $$x \to -\infty$$, and $$f(-3.331) > 0$$: one crossing in $$(-\infty, -3.331)$$.

2. $$f(-3.331) > 0$$ and $$f(-0.949) < 0$$: one crossing in $$(-3.331, -0.949)$$.

3. $$f(-0.949) < 0$$ and $$f(0.949) > 0$$: one crossing in $$(-0.949, 0.949)$$.

4. $$f(0.949) > 0$$ and $$f(3.331) < 0$$: one crossing in $$(0.949, 3.331)$$.

5. $$f(3.331) < 0$$ and $$f(x) \to +\infty$$ as $$x \to +\infty$$: one crossing in $$(3.331, +\infty)$$.

Each interval contains exactly one root (since $$f'$$ doesn't change sign within any interval between consecutive critical points), and each root is a simple root ($$f'$$ is nonzero there), so the curve truly crosses the x-axis at each.

The answer is $$5$$.

Solution :

Given function :

$$f(x)=\frac{5x^2}{2}+\frac{\alpha}{x^5}, \quad x>0$$

For minimum value :

$$f'(x)=0$$

Differentiate :

$$f'(x)=5x-\frac{5\alpha}{x^6}$$

Setting derivative equal to zero :

$$5x-\frac{5\alpha}{x^6}=0$$

$$5x=\frac{5\alpha}{x^6}$$

$$x^7=\alpha$$

$$x=\alpha^{1/7}$$

Second derivative :

$$f''(x)=5+\frac{30\alpha}{x^7}$$

Since for $$x>0$$ and $$\alpha>0$$,

$$f''(x)>0$$

therefore the function has minimum at :

$$x=\alpha^{1/7}$$

Now substitute in $$f(x)$$ :

$$f(\alpha^{1/7})=\frac{5(\alpha^{1/7})^2}{2}+\frac{\alpha}{(\alpha^{1/7})^5}$$

$$=\frac{5\alpha^{2/7}}{2}+\alpha^{2/7}$$

$$=\alpha^{2/7}\left(\frac{5}{2}+1\right)$$

$$=\frac{7\alpha^{2/7}}{2}$$

Given minimum value is :

$$14$$

Therefore,

$$\frac{7\alpha^{2/7}}{2}=14$$

$$\alpha^{2/7}=4$$

$$\alpha=4^{7/2}$$

$$=(2^2)^{7/2}$$

$$=2^7$$

$$=128$$

Final Answer :

$$128$$

We are given $$x, y > 0$$ with the constraint $$x^3 y^2 = 2^{15}$$, and we need to find the least value of $$3x + 2y$$.

Since the expression involves five terms, we apply the AM-GM inequality by writing $$ 3x + 2y = x + x + x + y + y $$ and then using $$ \frac{x + x + x + y + y}{5} \ge (x \cdot x \cdot x \cdot y \cdot y)^{1/5}. $$

From this we obtain $$ \frac{3x + 2y}{5} \ge (x^3 y^2)^{1/5}. $$ Substituting $$x^3 y^2 = 2^{15}$$ gives $$ \frac{3x + 2y}{5} \ge (2^{15})^{1/5} = 2^3 = 8, $$ so $$ 3x + 2y \ge 40. $$

Equality holds when all five terms are equal, which requires $$x = y$$. In that case, $$ x^3 \cdot x^2 = x^5 = 2^{15} \implies x = 2^3 = 8, $$ so $$x = y = 8$$ and $$ 3(8) + 2(8) = 24 + 16 = 40, $$ confirming the minimum value.

Therefore, the least value of $$3x + 2y$$ is 40, and the correct option is Option D.

The curve is $$y = x^3 + 3x^2 + 5$$, so $$\dfrac{dy}{dx} = 3x^2 + 6x$$.

The tangent at $$(x_1, y_1)$$ is: $$y - y_1 = (3x_1^2 + 6x_1)(x - x_1)$$.

Since it passes through the origin $$(0, 0)$$: $$-y_1 = (3x_1^2 + 6x_1)(0 - x_1) = -x_1(3x_1^2 + 6x_1)$$.

So $$y_1 = 3x_1^3 + 6x_1^2$$ $$-(1)$$

Since $$(x_1, y_1)$$ lies on the curve: $$y_1 = x_1^3 + 3x_1^2 + 5$$ $$-(2)$$

From $$(1)$$ and $$(2)$$: $$x_1^3 + 3x_1^2 + 5 = 3x_1^3 + 6x_1^2$$

$$2x_1^3 + 3x_1^2 - 5 = 0$$

We can factor this as $$(x_1 - 1)(2x_1^2 + 5x_1 + 5) = 0$$.

The quadratic $$2x_1^2 + 5x_1 + 5 = 0$$ has discriminant $$25 - 40 = -15 < 0$$, so the only real solution is $$x_1 = 1$$.

$$y_1 = 1 + 3 + 5 = 9$$. So the point is $$(1, 9)$$.

Now we check each option:

Option A: $$x^2 + \dfrac{y^2}{81} = 1 + \dfrac{81}{81} = 1 + 1 = 2$$ ✓ The point lies on this curve.

Option B: $$\dfrac{y^2}{9} - x^2 = \dfrac{81}{9} - 1 = 9 - 1 = 8$$ ✓ The point lies on this curve.

Option C: $$y = 4x^2 + 5 = 4(1) + 5 = 9$$ ✓ The point lies on this curve.

Option D: $$\dfrac{x}{3} - y^2 = \dfrac{1}{3} - 81 = -\dfrac{242}{3} \neq 0$$ ✗ The point does NOT lie on this curve.

The answer is Option D: $$\dfrac{x}{3} - y^2 = 0$$.

Given,

$$f(x)=(x^2-2x+7)e^{(4x^3-12x^2-180x+31)}$$

on the interval

$$[-3,0]$$

Let

$$g(x)=x^2-2x+7$$

and

$$h(x)=4x^3-12x^2-180x+31$$

Then,

$$f(x)=g(x)e^{h(x)}$$

Differentiate:

$$f'(x)=e^{h(x)}\left[g'(x)+g(x)h'(x)\right]$$

Now,

$$g'(x)=2x-2$$

and

$$h'(x)=12x^2-24x-180$$

$$=12(x-5)(x+3)$$

Therefore,

$$f'(x)=e^{h(x)}\left[(2x-2)+(x^2-2x+7)(12x^2-24x-180)\right]$$

Since

$$e^{h(x)}>0,$$

critical points come from

$$g'(x)+g(x)h'(x)=0$$

Now check important points in

$$[-3,0]$$

At

$$x=-3,$$

$$h(-3)=4(-27)-12(9)-180(-3)+31$$

$$=-108-108+540+31$$

$$=355$$

and

$$g(-3)=9+6+7=22$$

Hence,

$$f(-3)=22e^{355}$$

At

$$x=0,$$

$$h(0)=31$$

and

$$g(0)=7$$

Thus,

$$f(0)=7e^{31}$$

Clearly,

$$22e^{355}\gg7e^{31}$$

Also,

$$h'(x)=12(x-5)(x+3)<0$$

for

$$-3<x<0$$

Hence exponential part decreases throughout the interval, making

$$f(x)$$

strictly decreasing on

$$[-3,0]$$

Therefore, absolute maximum occurs at

$$x=-3$$

Hence,

$$\boxed{\alpha=-3}$$

We have $$f(x) = xe^{x(1-x)}$$. To find where this function is increasing or decreasing, we compute the derivative.

Since $$e^{x(1-x)} > 0$$ always, the sign of $$f'(x)$$ is determined by $$-(2x+1)(x-1)$$.

The critical points are $$x = -\frac{1}{2}$$ and $$x = 1$$.

For $$x < -\frac{1}{2}$$: $$(2x+1) < 0$$ and $$(x-1) < 0$$, so $$(2x+1)(x-1) > 0$$, hence $$f'(x) < 0$$ (decreasing).

For $$-\frac{1}{2} < x < 1$$: $$(2x+1) > 0$$ and $$(x-1) < 0$$, so $$(2x+1)(x-1) < 0$$, hence $$f'(x) > 0$$ (increasing).

For $$x > 1$$: $$(2x+1) > 0$$ and $$(x-1) > 0$$, so $$(2x+1)(x-1) > 0$$, hence $$f'(x) < 0$$ (decreasing).

So the function is increasing on $$\left(-\frac{1}{2}, 1\right)$$.

Checking the options: Option 1 says increasing in $$\left(-\frac{1}{2}, 1\right)$$, which matches perfectly.

Hence, the correct answer is Option 1.

Given,

$$x^4-4x+1=0$$

Let

$$f(x)=x^4-4x+1$$

Now,

$$f'(x)=4x^3-4=4(x^3-1)$$

Critical point is

$$x=1$$

Also,

$$f''(x)=12x^2>0$$

Hence, $$x=1$$ gives a minimum value.

Now,

$$f(1)=1-4+1=-2<0$$

Also,

$$\lim_{x\to\pm\infty}f(x)=+\infty$$

Further,

$$f(0)=1>0,\qquad f(2)=16-8+1=9>0$$

So, one root lies in $$(0,1)$$ and another root lies in $$(1,2)$$.

Also,

$$f(-1)=1+4+1=6>0$$

and the function is decreasing for $$x<1$$.

Hence, there is no negative root.

Therefore, the equation has exactly $$\boxed{2}$$ distinct real roots.

We have $$f(x) = 3(x^2 - 2)^3 + 4$$. We compute $$f'(x) = 3 \cdot 3(x^2 - 2)^2 \cdot 2x = 18x(x^2 - 2)^2$$.

Setting $$f'(x) = 0$$: either $$x = 0$$ or $$x^2 - 2 = 0$$, giving $$x = 0, \pm\sqrt{2}$$.

Statement P: $$x = 0$$ is a point of local minima. We observe that $$(x^2 - 2)^2 \geq 0$$ always, so the sign of $$f'(x)$$ is determined by the sign of $$x$$. For $$x$$ slightly less than 0, $$f'(x) < 0$$ (decreasing), and for $$x$$ slightly greater than 0, $$f'(x) > 0$$ (increasing). Since $$f'$$ changes sign from negative to positive, $$x = 0$$ is indeed a local minimum. P is true.

Statement Q: $$x = \sqrt{2}$$ is a point of inflection. We compute $$f''(x)$$. Differentiating $$f'(x) = 18x(x^2 - 2)^2$$: $$f''(x) = 18(x^2 - 2)^2 + 18x \cdot 2(x^2 - 2) \cdot 2x = 18(x^2 - 2)\big[(x^2 - 2) + 4x^2\big] = 18(x^2 - 2)(5x^2 - 2)$$.

At $$x = \sqrt{2}$$: $$f''(\sqrt{2}) = 18(2 - 2)(10 - 2) = 0$$. We check for a sign change: for $$x$$ slightly less than $$\sqrt{2}$$, $$x^2 - 2 < 0$$ while $$5x^2 - 2 > 0$$, so $$f'' < 0$$. For $$x$$ slightly greater than $$\sqrt{2}$$, both factors are positive, so $$f'' > 0$$. Since $$f''$$ changes sign, $$x = \sqrt{2}$$ is a point of inflection. Q is true.

Statement R: $$f'$$ is increasing for $$x > \sqrt{2}$$. This requires $$f''(x) > 0$$ for $$x > \sqrt{2}$$. We have $$f''(x) = 18(x^2 - 2)(5x^2 - 2)$$. For $$x > \sqrt{2}$$: $$x^2 > 2$$ so $$x^2 - 2 > 0$$, and $$5x^2 > 10 > 2$$ so $$5x^2 - 2 > 0$$. Hence $$f''(x) > 0$$, confirming $$f'$$ is increasing. R is true.

Since all three statements P, Q, and R are true, the correct answer is Option D.

Hence, the correct answer is Option D.

The curve $$ y(x) = ax^3 + bx^2 + cx + 5 $$ touches the $$ x $$-axis at $$ P(-2, 0) $$ and cuts the $$ y $$-axis at $$ Q $$ where $$ y' = 3 $$. We need to find the local maximum value.

Since the curve touches the $$ x $$-axis at $$ P(-2, 0) $$, both $$ y(-2) = 0 $$ and $$ y'(-2) = 0 $$.

At the $$ y $$-axis ($$ x = 0 $$): $$ y(0) = 5 $$ (point Q) and $$ y'(0) = 3 $$.

$$y'(x) = 3ax^2 + 2bx + c$$

From $$ y'(0) = 3 $$: $$ c = 3 $$.

From $$ y(-2) = 0 $$:

$$a(-8) + b(4) + c(-2) + 5 = 0$$

$$-8a + 4b - 6 + 5 = 0$$

$$-8a + 4b = 1 \quad \cdots (1)$$

From $$ y'(-2) = 0 $$:

$$3a(4) + 2b(-2) + 3 = 0$$

$$12a - 4b + 3 = 0$$

$$12a - 4b = -3 \quad \cdots (2)$$

Adding (1) and (2):

$$4a = -2 \implies a = -\frac{1}{2}$$

From (1): $$ -8(-\frac{1}{2}) + 4b = 1 \implies 4 + 4b = 1 \implies b = -\frac{3}{4} $$

So $$ y(x) = -\frac{1}{2}x^3 - \frac{3}{4}x^2 + 3x + 5 $$.

$$y'(x) = -\frac{3}{2}x^2 - \frac{3}{2}x + 3 = -\frac{3}{2}(x^2 + x - 2) = -\frac{3}{2}(x + 2)(x - 1)$$

$$ y'(x) = 0 $$ at $$ x = -2 $$ and $$ x = 1 $$.

$$y''(x) = -3x - \frac{3}{2}$$

At $$ x = -2 $$: $$ y''(-2) = 6 - \frac{3}{2} = \frac{9}{2} > 0 $$ — local minimum.

At $$ x = 1 $$: $$ y''(1) = -3 - \frac{3}{2} = -\frac{9}{2} < 0 $$ — local maximum.

$$y(1) = -\frac{1}{2}(1) - \frac{3}{4}(1) + 3(1) + 5 = -\frac{1}{2} - \frac{3}{4} + 3 + 5$$

$$= -\frac{2}{4} - \frac{3}{4} + 8 = -\frac{5}{4} + 8 = \frac{-5 + 32}{4} = \frac{27}{4}$$

The local maximum value is $$ \dfrac{27}{4} $$, which corresponds to Option A.

The sides of the triangle are

$$10+x^2,\qquad 10+x^2,\qquad 20-2x^2$$

Hence, the triangle is isosceles.

The semi-perimeter is

$$s=\frac{(10+x^2)+(10+x^2)+(20-2x^2)}{2}=20$$

Using Heron's formula,

$$\Delta^2=s(s-a)(s-b)(s-c)$$

$$=20(10-x^2)(10-x^2)(2x^2)$$

$$=40x^2(10-x^2)^2$$

Therefore,

$$\Delta=2\sqrt{10}\,x(10-x^2)$$

To maximize area, maximize

$$f(x)=10x-x^3$$

Now,

$$f'(x)=10-3x^2$$

For maximum area,

$$10-3x^2=0$$

$$3x^2=10$$

Since maximum occurs at

$$x=k$$

we get

$$3k^2=10$$

Hence,

$$\boxed{10}$$

We need to find the number of distinct real roots of $$x^7 - 7x - 2 = 0$$.

Let $$f(x) = x^7 - 7x - 2$$.

Finding critical points: $$f'(x) = 7x^6 - 7 = 7(x^6 - 1) = 0$$

This gives $$x^6 = 1$$, so $$x = 1$$ and $$x = -1$$ are the only real critical points.

$$f''(x) = 42x^5$$

At $$x = -1$$: $$f''(-1) = -42 < 0$$, so $$x = -1$$ is a local maximum.

At $$x = 1$$: $$f''(1) = 42 > 0$$, so $$x = 1$$ is a local minimum.

Evaluating $$f$$ at the critical points:

$$f(-1) = (-1)^7 - 7(-1) - 2 = -1 + 7 - 2 = 4 > 0$$

$$f(1) = 1 - 7 - 2 = -8 < 0$$

Since $$f$$ is a degree 7 polynomial with positive leading coefficient:

As $$x \to -\infty$$, $$f(x) \to -\infty$$ and as $$x \to +\infty$$, $$f(x) \to +\infty$$.

Analyzing sign changes:

From $$-\infty$$ to $$x = -1$$: $$f$$ goes from $$-\infty$$ to $$f(-1) = 4 > 0$$. So there is exactly one root in $$(-\infty, -1)$$.

From $$x = -1$$ to $$x = 1$$: $$f$$ goes from $$4 > 0$$ to $$f(1) = -8 < 0$$. So there is exactly one root in $$(-1, 1)$$.

From $$x = 1$$ to $$+\infty$$: $$f$$ goes from $$-8 < 0$$ to $$+\infty$$. So there is exactly one root in $$(1, +\infty)$$.

Since the function is monotonically decreasing between the local max and local min, and monotonically increasing outside, there are exactly 3 distinct real roots.

The correct answer is Option D.

We need to find the number of real solutions of $$x^7 + 5x^3 + 3x + 1 = 0$$. Let $$f(x) = x^7 + 5x^3 + 3x + 1$$, so $$f'(x) = 7x^6 + 15x^2 + 3$$. Since $$7x^6 \geq 0$$ and $$15x^2 \geq 0$$ for all real $$x$$ and $$3 > 0$$, it follows that $$f'(x) \geq 3 > 0$$ for all $$x \in \mathbb{R}$$, and hence $$f(x)$$ is strictly increasing on $$\mathbb{R}$$.

As $$x \to -\infty$$, $$f(x) \to -\infty$$ (dominated by $$x^7$$), and as $$x \to +\infty$$, $$f(x) \to +\infty$$. Since $$f$$ is continuous and strictly increasing, by the Intermediate Value Theorem it crosses the $$x$$-axis exactly once.

The equation $$x^7 + 5x^3 + 3x + 1 = 0$$ has exactly 1 real solution. Therefore, the correct answer is Option B: $$1$$.

We have $$f(x) = \tan^{-1}(\sin x - \cos x)$$ on $$[0, \pi]$$. Since $$\tan^{-1}$$ is an increasing function, the extrema of $$f(x)$$ correspond to the extrema of $$g(x) = \sin x - \cos x$$.

Now $$g(x) = \sin x - \cos x = \sqrt{2}\sin\left(x - \frac{\pi}{4}\right)$$.

On $$[0, \pi]$$, the argument $$x - \frac{\pi}{4}$$ ranges over $$\left[-\frac{\pi}{4}, \frac{3\pi}{4}\right]$$.

The maximum of $$\sin\left(x - \frac{\pi}{4}\right)$$ on this interval is 1, occurring at $$x - \frac{\pi}{4} = \frac{\pi}{2}$$, i.e., $$x = \frac{3\pi}{4}$$. So the maximum of $$g$$ is $$\sqrt{2}$$.

The minimum of $$\sin\left(x - \frac{\pi}{4}\right)$$ on $$\left[-\frac{\pi}{4}, \frac{3\pi}{4}\right]$$ is $$\sin\left(-\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}$$, occurring at $$x = 0$$. So the minimum of $$g$$ is $$\sqrt{2} \cdot \left(-\frac{\sqrt{2}}{2}\right) = -1$$.

Therefore:

Absolute maximum of $$f = \tan^{-1}(\sqrt{2})$$

Absolute minimum of $$f = \tan^{-1}(-1) = -\frac{\pi}{4}$$

The sum is $$\tan^{-1}(\sqrt{2}) - \frac{\pi}{4}$$.

Now we need to check if $$\tan^{-1}(\sqrt{2}) = \cos^{-1}\left(\frac{1}{\sqrt{3}}\right)$$. If $$\theta = \tan^{-1}(\sqrt{2})$$, then $$\tan\theta = \sqrt{2}$$, so $$\sec^2\theta = 1 + 2 = 3$$, giving $$\cos^2\theta = \frac{1}{3}$$, hence $$\cos\theta = \frac{1}{\sqrt{3}}$$ (since $$\theta$$ is in the first quadrant). Therefore $$\theta = \cos^{-1}\left(\frac{1}{\sqrt{3}}\right)$$.

So the sum equals $$\cos^{-1}\left(\frac{1}{\sqrt{3}}\right) - \frac{\pi}{4}$$.

Hence, the correct answer is Option 3.

We need to find the ratio $$x : r$$ that maximizes the sum of volumes of a cuboid with sides $$2x, 4x, 5x$$ and a closed hemisphere of radius $$r$$, given that the sum of their surface areas is a constant $$k$$.

First, we write the surface area constraint. The surface area of the cuboid is $$2(2x \cdot 4x + 4x \cdot 5x + 2x \cdot 5x) = 2(8x^2 + 20x^2 + 10x^2) = 76x^2$$, and the surface area of the closed hemisphere (curved surface + flat circular base) is $$2\pi r^2 + \pi r^2 = 3\pi r^2$$. Since the total surface area is constant, we have $$76x^2 + 3\pi r^2 = k$$.

Next, we write the total volume. The volume of the cuboid is $$2x \cdot 4x \cdot 5x$$, and the volume of the hemisphere is $$\frac{2}{3}\pi r^3$$, so that the combined volume is $$V = 40x^3 + \frac{2}{3}\pi r^3$$.

Now, we differentiate the constraint with respect to $$x$$ in order to relate $$\frac{dr}{dx}$$. Differentiating $$76x^2 + 3\pi r^2 = k$$ gives $$152x + 6\pi r \frac{dr}{dx} = 0$$, which implies $$\frac{dr}{dx} = -\frac{152x}{6\pi r} = -\frac{76x}{3\pi r}$$.

Then, we set $$\frac{dV}{dx} = 0$$ for maximum volume. Since $$\frac{dV}{dx} = 120x^2 + 2\pi r^2 \cdot \frac{dr}{dx}$$, substituting the expression for $$\frac{dr}{dx}$$ yields $$120x^2 + 2\pi r^2 \left(-\frac{76x}{3\pi r}\right) = 0$$, which simplifies to $$120x^2 - \frac{152xr}{3} = 0$$. Therefore, $$120x = \frac{152r}{3}$$, so that $$360x = 152r$$ and hence $$\frac{x}{r} = \frac{152}{360} = \frac{19}{45}$$.

Subsequently, we conclude that $$x : r = 19 : 45$$, and thus the answer is Option B: $$19 : 45$$.

We are given $$f(x) = 4\log_e(x-1) - 2x^2 + 4x + 5$$ for $$x > 1$$.

Find f'(x) and critical points.

$$f'(x) = \frac{4}{x-1} - 4x + 4 = \frac{4}{x-1} - 4(x-1) = \frac{4 - 4(x-1)^2}{x-1}$$

Setting $$f'(x) = 0$$: $$4 - 4(x-1)^2 = 0 \implies (x-1)^2 = 1 \implies x = 2$$ (since $$x > 1$$).

For $$x \in (1, 2)$$, $$(x-1)^2 < 1$$, so $$f'(x) > 0$$ (increasing).

For $$x \in (2, \infty)$$, $$(x-1)^2 > 1$$, so $$f'(x) < 0$$ (decreasing).

So Option A is correct.

Find f(2) and check Option B.

$$f(2) = 4\ln(1) - 8 + 8 + 5 = 5$$

Since $$f$$ increases on $$(1,2)$$ and decreases on $$(2,\infty)$$, the maximum is $$f(2) = 5$$. As $$x \to 1^+$$, $$f(x) \to -\infty$$, and as $$x \to \infty$$, $$f(x) \to -\infty$$. So $$f(x) = -1$$ (which is less than 5) has exactly two solutions (one on each side of $$x = 2$$). Option B is correct.

Compute f'(e) and f''(2) to check Option C.

$$f'(e) = \frac{4}{e-1} - 4(e-1) = \frac{4 - 4(e-1)^2}{e-1}$$

Since $$(e-1)^2 \approx (1.718)^2 \approx 2.952$$, we get $$f'(e) = \frac{4 - 11.81}{1.718} \approx -4.55$$.

$$f''(x) = -\frac{4}{(x-1)^2} - 4$$

$$f''(2) = -\frac{4}{1} - 4 = -8$$

$$f'(e) - f''(2) \approx -4.55 - (-8) = 3.45 > 0$$

Option C claims $$f'(e) - f''(2) < 0$$, which is false.

Check Option D.

$$f(e) = 4\ln(e-1) - 2e^2 + 4e + 5 = 4\ln(1.718) - 2(7.389) + 4(2.718) + 5 \approx 2.164 - 14.778 + 10.873 + 5 = 3.26 > 0$$

$$f(e+1) = 4\ln(e) - 2(e+1)^2 + 4(e+1) + 5 = 4 - 2(13.87) + 14.87 + 5 \approx -3.87 < 0$$

By the Intermediate Value Theorem, $$f(x) = 0$$ has a root in $$(e, e+1)$$. Option D is correct.

Answer: Option C is NOT correct.

Given,

$$f(x)=\int_0^{x^2}\frac{t^2-5t+4}{2+e^t}\,dt$$

Using Leibnitz rule,

$$f'(x)=\frac{x^4-5x^2+4}{2+e^{x^2}}\cdot2x$$

$$=\frac{2x(x^2-1)(x^2-4)}{2+e^{x^2}}$$

Since,

$$2+e^{x^2}>0$$

for all $$x$$, the critical points are obtained from

$$2x(x^2-1)(x^2-4)=0$$

Hence,

$$x=0,\pm1,\pm2$$

Now, checking sign changes of

$$x(x^2-1)(x^2-4)$$

we get:

- Increasing on

$$(-\infty,-2),\ (-1,0),\ (1,2)$$

- Decreasing on

$$(-2,-1),\ (0,1),\ (2,\infty)$$

Therefore,

- Local maxima occur at

$$x=-2,\ 0,\ 2$$

So,

$$m=3$$

- Local minima occur at

$$x=-1,\ 1$$

So,

$$n=2$$

Hence,

$$\boxed{(m,n)=(3,2)}$$

Given,

$$f(x)=\log_e(x^2+1)-e^{-x}+1$$

and

$$g(x)=\frac{1-2e^{2x}}{e^x}=e^{-x}-2e^x$$

We need to solve

$$f\left(g\left(\frac{(\alpha-1)^2}{3}\right)\right)>f\left(g\left(\alpha-\frac53\right)\right)$$

First, find the nature of $$f$$.

Differentiating,

$$f'(x)=\frac{2x}{x^2+1}+e^{-x}$$

Now,

$$\frac{2x}{x^2+1}\ge -1$$

and

$$e^{-x}>0$$

Hence,

$$f'(x)>0\qquad \forall x\in\mathbb R$$

Therefore, $$f(x)$$ is strictly increasing.

So,

$$f(A)>f(B)\iff A>B$$

Hence, the given inequality becomes

$$g\left(\frac{(\alpha-1)^2}{3}\right)>g\left(\alpha-\frac53\right)$$

Now,

$$g(x)=e^{-x}-2e^x$$

Differentiating,

$$g'(x)=-e^{-x}-2e^x<0$$

Therefore, $$g(x)$$ is strictly decreasing.

Hence,

$$g(u)>g(v)\iff u<v$$

So,

$$\frac{(\alpha-1)^2}{3}<\alpha-\frac53$$

Multiplying by $$3,$$

$$ (\alpha-1)^2<3\alpha-5 $$

$$ \alpha^2-2\alpha+1<3\alpha-5 $$

$$ \alpha^2-5\alpha+6<0 $$

$$ (\alpha-2)(\alpha-3)<0 $$

Therefore,

$$2<\alpha<3$$

Hence, the required range of $$\alpha$$ is $$\boxed{(2,3)}$$.

Given $$f(x) = (x - 3)^{n_1}(x - 5)^{n_2}$$ where $$n_1, n_2 \in \mathbb{N}$$. We need to determine which statement about local maxima in $$(3, 5)$$ is NOT true.

Finding the critical point: Differentiating:

$$f'(x) = (x-3)^{n_1-1}(x-5)^{n_2-1}\left[n_1(x-5) + n_2(x-3)\right]$$

Setting the linear factor to zero: $$n_1(x-5) + n_2(x-3) = 0$$ gives $$\alpha = \dfrac{5n_1 + 3n_2}{n_1 + n_2}$$, which lies in $$(3, 5)$$ for all positive $$n_1, n_2$$.

Nature of the critical point: Since $$f(3) = f(5) = 0$$, if $$f(x) > 0$$ in $$(3, 5)$$, then $$f$$ has a local maximum; if $$f(x) < 0$$ in $$(3, 5)$$, then $$f$$ has a local minimum.

In $$(3, 5)$$: $$(x-3)^{n_1} > 0$$ always (since $$x - 3 > 0$$). The sign of $$(x-5)^{n_2}$$ depends on the parity of $$n_2$$ (since $$x - 5 < 0$$).

• $$n_2$$ even $$\Rightarrow$$ $$(x-5)^{n_2} > 0$$ $$\Rightarrow$$ $$f(x) > 0$$ $$\Rightarrow$$ local maximum.
• $$n_2$$ odd $$\Rightarrow$$ $$(x-5)^{n_2} < 0$$ $$\Rightarrow$$ $$f(x) < 0$$ $$\Rightarrow$$ local minimum.

Checking Option C ($$n_1 = 3, n_2 = 5$$): Since $$n_2 = 5$$ is odd, $$f(x) < 0$$ throughout $$(3, 5)$$ while $$f(3) = f(5) = 0$$. The function dips below zero and returns, giving a local minimum at $$\alpha$$, not a local maximum.

We can verify using the first derivative test at $$\alpha = \dfrac{5(3)+3(5)}{3+5} = \dfrac{30}{8} = 3.75$$:

$$f'(x) = (x-3)^2(x-5)^4(8x - 30)$$

For $$x < 3.75$$: $$(x-3)^2 > 0$$, $$(x-5)^4 > 0$$, $$(8x-30) < 0$$ $$\Rightarrow$$ $$f'(x) < 0$$.

For $$x > 3.75$$: $$(8x-30) > 0$$ $$\Rightarrow$$ $$f'(x) > 0$$.

Since $$f'$$ changes from negative to positive, $$\alpha = 3.75$$ is a point of local minimum, confirming that Option C's claim of local maxima is false.

The correct answer is Option C.

We need to find all values of $$b$$ for which $$f(x)$$ has a maximum value at $$x = 1$$.

Using the first piece since $$x \le 1$$ gives $$f(1) = 1 - 1 + 10 - 7 = 3$$.

Next, for $$x \le 1$$ the function is $$f(x) = x^3 - x^2 + 10x - 7$$ and $$f'(x) = 3x^2 - 2x + 10$$. Since the discriminant of $$3x^2 - 2x + 10$$ is $$4 - 120 = -116 < 0$$ and the leading coefficient is positive, $$f'(x) > 0$$ on $$(-\infty,1]$$. This means $$f$$ is strictly increasing on $$(-\infty,1]$$, so $$f(x) \le f(1) = 3$$ for all $$x \le 1$$.

For $$x > 1$$, $$f(x) = -2x + \log_2(b^2 - 4)$$. Since the coefficient of $$x$$ is $$-2 < 0$$, this expression is strictly decreasing on $$(1,\infty)$$, so its supremum occurs as $$x \to 1^+$$: $$\lim_{x \to 1^+} f(x) = -2 + \log_2(b^2 - 4)$$.

To ensure a maximum at $$x = 1$$, we require $$f(x) \le 3$$ for all $$x > 1$$, which gives $$-2 + \log_2(b^2 - 4) \le 3$$, then $$\log_2(b^2 - 4) \le 5$$, so $$b^2 - 4 \le 32$$, hence $$b^2 \le 36$$ and $$-6 \le b \le 6$$.

Moreover, for $$\log_2(b^2 - 4)$$ to be defined we require $$b^2 - 4 > 0$$, i.e., $$b < -2$$ or $$b > 2$$. Combining this with $$-6 \le b \le 6$$ gives $$b \in [-6, -2) \cup (2, 6]$$.

The correct answer is Option C: $$[-6, -2) \cup (2, 6]$$.

We are given $$f_\lambda(x) = 4\lambda x^3 - 36\lambda x^2 + 36x + 48$$ and need to find the largest $$\lambda$$ for which $$f_\lambda$$ is increasing for all $$x \in \mathbb{R}$$.

For $$f_\lambda$$ to be increasing for all $$x$$, we need $$f'_\lambda(x) \geq 0$$ for all $$x$$.

$$f'_\lambda(x) = 12\lambda x^2 - 72\lambda x + 36$$

If $$\lambda = 0$$: $$f'(x) = 36 > 0$$ for all $$x$$, so $$\lambda = 0$$ works. But we want the largest $$\lambda$$.

If $$\lambda < 0$$: The quadratic $$12\lambda x^2 - 72\lambda x + 36$$ opens downward (since $$12\lambda < 0$$), so it will eventually become negative. Hence $$\lambda < 0$$ does not work.

If $$\lambda > 0$$: The quadratic opens upward. For it to be non-negative for all $$x$$, we need the discriminant $$\leq 0$$.

Discriminant $$= (72\lambda)^2 - 4(12\lambda)(36) = 5184\lambda^2 - 1728\lambda$$

Setting discriminant $$\leq 0$$:

$$5184\lambda^2 - 1728\lambda \leq 0$$

$$\lambda(5184\lambda - 1728) \leq 0$$

$$\lambda(3\lambda - 1) \leq 0$$ (dividing by 1728)

Since $$\lambda > 0$$, we need $$3\lambda - 1 \leq 0$$, i.e., $$\lambda \leq \frac{1}{3}$$.

Therefore $$\lambda^* = \frac{1}{3}$$.

Now computing $$f_{1/3}(1) + f_{1/3}(-1)$$:

$$f_{1/3}(x) = \frac{4}{3}x^3 - 12x^2 + 36x + 48$$

$$f_{1/3}(1) = \frac{4}{3} - 12 + 36 + 48 = \frac{4}{3} + 72 = \frac{220}{3}$$

$$f_{1/3}(-1) = -\frac{4}{3} - 12 - 36 + 48 = -\frac{4}{3} + 0 = -\frac{4}{3}$$

$$f_{1/3}(1) + f_{1/3}(-1) = \frac{220}{3} - \frac{4}{3} = \frac{216}{3} = 72$$

The correct answer is Option D.

Let $$Q(t,t^2)$$ be a point on the parabola $$y=x^2.$$

The circle $$(x-1)^2+(y+1)^2=1$$ has centre $$C(1,-1)$$ and radius $$1.$$

For minimum distance between the circle and parabola, we minimize the distance between $$C(1,-1)$$ and $$Q(t,t^2).$$

Let

$$D^2=(t-1)^2+(t^2+1)^2$$

$$=t^2-2t+1+t^4+2t^2+1$$

$$=t^4+3t^2-2t+2$$

Now differentiate:

$$\frac{d(D^2)}{dt}=4t^3+6t-2$$

For minimum distance,

$$4t^3+6t-2=0$$

$$2t^3+3t-1=0$$

Checking signs,

$$f\left(\frac14\right)=2\left(\frac1{64}\right)+\frac34-1<0$$

and

$$f\left(\frac12\right)=2\left(\frac18\right)+\frac32-1>0$$

Hence,

$$t\in\left(\frac14,\frac12\right)$$

Now point $$P$$ lies on the circle along the line joining $$C$$ and $$Q.$$

Hence,

$$P=C+\frac{Q-C}{|Q-C|}$$

Therefore, abscissa of $$P$$ is

$$x_P=1+\frac{t-1}{\sqrt{(t-1)^2+(t^2+1)^2}}$$

From

$$2t^3+3t-1=0,$$

$$1-t=2t(t^2+1)$$

Substituting,

$$x_P=1-\frac{2t(t^2+1)}{\sqrt{4t^2(t^2+1)^2+(t^2+1)^2}}$$

$$=1-\frac{2t}{\sqrt{4t^2+1}}$$

Now for

$$t\in\left(\frac14,\frac12\right),$$

$$x_P$$

lies between

$$1-\frac1{\sqrt5}$$

and

$$1-\frac1{\sqrt2}$$

Numerically,

$$0.29<x_P<0.55$$

Hence, the abscissa of $$P$$ lies in the interval

$$\boxed{\left(\frac14,\frac12\right)}$$.

We need to find the sum of the absolute minimum and the absolute maximum values of $$f(x) = |3x - x^2 + 2| - x$$ on $$[-1, 2]$$. First, determine where the expression inside the absolute value changes sign. Setting $$-x^2 + 3x + 2 = 0$$ gives $$x^2 - 3x - 2 = 0$$, so $$x = \frac{3 \pm \sqrt{9 + 8}}{2} = \frac{3 \pm \sqrt{17}}{2}$$, and the relevant root in $$[-1, 2]$$ is $$r = \frac{3 - \sqrt{17}}{2} \approx -0.56$$. Since at $$x = 0$$ we have $$-0 + 0 + 2 = 2 > 0$$, it follows that $$-x^2 + 3x + 2 \geq 0$$ for $$x \in [r, 2]$$ and $$\leq 0$$ for $$x \in [-1, r]$$.

Accordingly, for $$x \in [-1, r]$$ we have $$f(x) = (x^2 - 3x - 2) - x = x^2 - 4x - 2$$, while for $$x \in [r, 2]$$ we have $$f(x) = (-x^2 + 3x + 2) - x = -x^2 + 2x + 2$$.

On the interval $$[-1, r]$$ the derivative is $$f'(x) = 2x - 4$$, which vanishes at $$x = 2$$, outside the interval, so there are no interior critical points in $$[-1, r]$$. On $$[r, 2]$$ the derivative is $$f'(x) = -2x + 2$$, which vanishes at $$x = 1$$, lying in $$[r, 2]$$.

Next, evaluate $$f(x)$$ at the boundary and critical points. At $$x = -1$$ one finds $$f(-1) = 1 + 4 - 2 = 3$$. At $$x = r$$, since $$r$$ satisfies $$r^2 - 3r - 2 = 0$$ so that $$r^2 = 3r + 2$$, we have $$f(r) = r^2 - 4r - 2 = (3r + 2) - 4r - 2 = -r = \frac{\sqrt{17} - 3}{2} \approx 0.56$$. At $$x = 1$$ we get $$f(1) = -1 + 2 + 2 = 3$$, and at $$x = 2$$ we obtain $$f(2) = -4 + 4 + 2 = 2$$.

Since the values are $$f(-1) = 3$$, $$f(r) = \frac{\sqrt{17}-3}{2}$$, $$f(1) = 3$$, and $$f(2) = 2$$, the absolute maximum is $$3$$ and the absolute minimum is $$\frac{\sqrt{17}-3}{2}$$. Therefore, their sum is $$3 + \frac{\sqrt{17} - 3}{2} = \frac{6 + \sqrt{17} - 3}{2} = \frac{\sqrt{17} + 3}{2}$$.

Thus, the correct answer is Option A: $$\frac{\sqrt{17}+3}{2}$$.

Since the conical vessel has height $$H = 35$$ cm and radius $$R = 7$$ cm, by similar triangles at water level height $$h$$ one obtains $$\frac{r}{h} = \frac{7}{35} = \frac{1}{5} \Rightarrow r = \frac{h}{5}$$.

The volume of water is given by $$V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi \left(\frac{h}{5}\right)^2 h = \frac{\pi h^3}{75}$$, so differentiating with respect to time yields $$\frac{dV}{dt} = \frac{\pi h^2}{25} \cdot \frac{dh}{dt} = 1$$. Substituting $$h = 10$$ gives $$\frac{\pi (100)}{25} \cdot \frac{dh}{dt} = 1 \Rightarrow 4\pi \cdot \frac{dh}{dt} = 1 \Rightarrow \frac{dh}{dt} = \frac{1}{4\pi}$$.

Next, the slant height satisfies $$l = \sqrt{r^2 + h^2} = \sqrt{\frac{h^2}{25} + h^2} = h\sqrt{\frac{26}{25}} = \frac{h\sqrt{26}}{5}$$, and hence the wet conical surface area is $$S = \pi r l = \pi \cdot \frac{h}{5} \cdot \frac{h\sqrt{26}}{5} = \frac{\pi h^2 \sqrt{26}}{25}$$.

Therefore, differentiating gives $$\frac{dS}{dt} = \frac{2\pi h \sqrt{26}}{25} \cdot \frac{dh}{dt}$$, and at $$h = 10$$ one finds $$\frac{dS}{dt} = \frac{2\pi \cdot 10 \cdot \sqrt{26}}{25} \cdot \frac{1}{4\pi} = \frac{20\pi\sqrt{26}}{25} \cdot \frac{1}{4\pi} = \frac{20\sqrt{26}}{100} = \frac{\sqrt{26}}{5}$$.

Therefore, the rate of increase of wet conical surface area is $$\frac{\sqrt{26}}{5}$$ cm$$^2$$/sec, which is Option C.

For

$$f_a(x)$$

to be non-decreasing on

$$\left(-\frac{\pi}{6},\frac{\pi}{6}\right)$$,

we must have

$$f_a'(x)\ge 0$$

throughout the interval.

Differentiating,

$$f_a'(x)=\frac{2}{1+4x^2}-3a$$

Therefore,

$$\frac{2}{1+4x^2}-3a\ge 0$$

$$3a\le \frac{2}{1+4x^2}$$

for every

$$x\in\left(-\frac{\pi}{6},\frac{\pi}{6}\right)$$.

To obtain the maximum possible value of

$$a$$,

we need the minimum value of

$$\frac{2}{1+4x^2}$$

on the interval.

Since

$$x^2$$

is largest at

$$x=\pm\frac{\pi}{6}$$,

we get

$$3\bar a=\frac{2}{1+\frac{\pi^2}{9}}$$

$$=\frac{18}{9+\pi^2}$$

Hence

$$\bar a=\frac{6}{9+\pi^2}$$

Now,

$$f_{\bar a}\left(\frac{\pi}{8}\right)=\tan^{-1}\left(\frac{\pi}{4}\right)-3\bar a\left(\frac{\pi}{8}\right)+7$$

Substituting

$$\bar a=\frac{6}{9+\pi^2}$$,

$$f_{\bar a}\left(\frac{\pi}{8}\right)=7+\tan^{-1}\left(\frac{\pi}{4}\right)-\frac{18}{9+\pi^2}\cdot\frac{\pi}{8}$$

$$=7+\tan^{-1}\left(\frac{\pi}{4}\right)-\frac{9\pi}{4(9+\pi^2)}$$

Therefore,

$$f_{\bar a}\left(\frac{\pi}{8}\right)=8-\frac{9\pi}{4(9+\pi^2)}$$

We are given the curve $$y = x^3 + 3x^2 + 5$$ and the tangent at $$(x_1, y_1)$$ passes through the origin.

Find the tangent equation.

The slope at $$(x_1, y_1)$$ is $$y' = 3x_1^2 + 6x_1$$.

The tangent line: $$y - y_1 = (3x_1^2 + 6x_1)(x - x_1)$$.

Apply the condition that the tangent passes through the origin.

Substituting $$(0, 0)$$: $$-y_1 = (3x_1^2 + 6x_1)(-x_1)$$

$$y_1 = 3x_1^3 + 6x_1^2$$

Use the fact that $$(x_1, y_1)$$ lies on the curve.

$$y_1 = x_1^3 + 3x_1^2 + 5$$

Equating: $$x_1^3 + 3x_1^2 + 5 = 3x_1^3 + 6x_1^2$$

$$2x_1^3 + 3x_1^2 - 5 = 0$$

Solve the cubic.

Testing $$x_1 = 1$$: $$2 + 3 - 5 = 0$$ ✓

Factoring: $$(x_1 - 1)(2x_1^2 + 5x_1 + 5) = 0$$

The discriminant of $$2x_1^2 + 5x_1 + 5$$ is $$25 - 40 = -15 < 0$$, so the only real solution is $$x_1 = 1$$.

$$y_1 = 1 + 3 + 5 = 9$$. So $$(x_1, y_1) = (1, 9)$$.

Check which curve (1, 9) does NOT lie on.

Option A: $$1 + \frac{81}{81} = 1 + 1 = 2$$ ✓

Option B: $$\frac{81}{9} - 1 = 9 - 1 = 8$$ ✓

Option C: $$4(1) + 5 = 9$$ ✓

Option D: $$\frac{1}{3} - 81 = -\frac{242}{3} \neq 2$$ ✗

Answer: Option D

We need to find the set $$S$$ of natural numbers $$n$$ for which the line $$\frac{x}{a} + \frac{y}{b} = 2$$ is tangent to the curve $$\left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2$$ at the point $$(a, b)$$.

At $$(a, b)$$ the curve gives $$1^n + 1^n = 2$$ $$\checkmark$$ and the line gives $$1 + 1 = 2$$ $$\checkmark$$.

Next, differentiate $$\left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2$$ implicitly to get
$$\frac{n}{a}\left(\frac{x}{a}\right)^{n-1} + \frac{n}{b}\left(\frac{y}{b}\right)^{n-1}\frac{dy}{dx} = 0$$ Substituting $$(x,y)=(a,b)$$ yields $$\frac{n}{a}(1) + \frac{n}{b}(1)\frac{dy}{dx} = 0$$, so that $$\frac{dy}{dx} = -\frac{b}{a}$$.

Then the tangent line at $$(a,b)$$ satisfies $$y - b = -\frac{b}{a}(x - a)$$. It follows that $$y = -\frac{b}{a}x + b + b = -\frac{b}{a}x + 2b$$, which leads to $$\frac{b}{a}x + y = 2b$$ and hence $$\frac{x}{a} + \frac{y}{b} = 2$$.

Thus the tangent at $$(a, b)$$ to the curve is $$\frac{x}{a} + \frac{y}{b} = 2$$ for every natural number $$n$$, since the slope $$-\frac{b}{a}$$ is independent of $$n$$. Therefore, the given line is tangent to the curve at $$(a, b)$$ for all $$n \in \mathbb{N}$$.

The correct answer is Option D: $$S = \mathbb{N}$$.

A wire of length 22 m is cut into two pieces. One piece forms a square and the other an equilateral triangle.

First, let the side of the equilateral triangle be $$a$$, so its perimeter is $$3a$$. The remaining wire for the square is $$22 - 3a$$, and hence the side of the square is $$\frac{22 - 3a}{4}$$.

Next, the area of the square is $$\left(\frac{22 - 3a}{4}\right)^2 = \frac{(22-3a)^2}{16}$$, while the area of the equilateral triangle is $$\frac{\sqrt{3}}{4}a^2$$. Therefore, the total area as a function of $$a$$ is $$A(a) = \frac{(22-3a)^2}{16} + \frac{\sqrt{3}}{4}a^2$$.

To minimize this area, we differentiate to obtain $$A'(a) = \frac{2(22-3a)(-3)}{16} + \frac{\sqrt{3}}{2}a = \frac{-3(22-3a)}{8} + \frac{\sqrt{3}}{2}a$$. Setting $$A'(a) = 0$$ gives $$\frac{3(22-3a)}{8} = \frac{\sqrt{3}}{2}a$$, which leads to $$3(22-3a) = 4\sqrt{3}\,a$$ and hence $$66 - 9a = 4\sqrt{3}\,a$$. Combining terms yields $$66 = a(9 + 4\sqrt{3})$$, so $$a = \frac{66}{9 + 4\sqrt{3}}$$.

Therefore, the length of the side of the equilateral triangle that minimizes the total area is $$\frac{66}{9+4\sqrt{3}}$$. The answer is Option B.

We are given the parametric curve $$x = 12(t + \sin t \cos t)$$ and $$y = 12(1 + \sin t)^2$$ where $$0 < t < \frac{\pi}{2}$$.

The slope of the tangent is $$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$$.

First, compute $$\frac{dx}{dt}$$:

$$\frac{dx}{dt} = 12(1 + \cos^2 t - \sin^2 t) = 12(1 + \cos 2t) = 24\cos^2 t$$

Next, compute $$\frac{dy}{dt}$$:

$$\frac{dy}{dt} = 12 \cdot 2(1 + \sin t) \cdot \cos t = 24(1 + \sin t)\cos t$$

Therefore:

$$\frac{dy}{dx} = \frac{24(1 + \sin t)\cos t}{24\cos^2 t} = \frac{1 + \sin t}{\cos t}$$

Since the tangent makes angle $$\frac{\pi}{3}$$ with the positive $$x$$-axis:

$$\frac{1 + \sin t}{\cos t} = \tan\frac{\pi}{3} = \sqrt{3}$$

So $$1 + \sin t = \sqrt{3}\cos t$$.

Rearranging: $$\sqrt{3}\cos t - \sin t = 1$$

This can be written as $$2\sin\left(\frac{\pi}{3} - t\right) = 1$$, giving $$\sin\left(\frac{\pi}{3} - t\right) = \frac{1}{2}$$.

Since $$0 < t < \frac{\pi}{2}$$, we get $$\frac{\pi}{3} - t = \frac{\pi}{6}$$, so $$t = \frac{\pi}{6}$$.

Now compute $$y_0$$:

$$y_0 = 12\left(1 + \sin\frac{\pi}{6}\right)^2 = 12\left(1 + \frac{1}{2}\right)^2 = 12 \cdot \frac{9}{4} = 27$$

The answer is Option C: $$27$$.

We need the sum of the absolute maximum and minimum of $$f(x) = |2x^2 + 3x - 2| + \sin x \cos x$$ on $$[0, 1]$$.

Analyze the expression inside the absolute value.

$$2x^2 + 3x - 2 = (2x - 1)(x + 2)$$

On $$[0, 1]$$: $$(x + 2) > 0$$ always. $$(2x - 1) < 0$$ for $$x \in [0, 1/2)$$ and $$(2x - 1) > 0$$ for $$x \in (1/2, 1]$$.

Write f(x) piecewise.

For $$x \in [0, 1/2]$$: $$f(x) = -(2x^2 + 3x - 2) + \frac{\sin 2x}{2} = -2x^2 - 3x + 2 + \frac{\sin 2x}{2}$$

For $$x \in [1/2, 1]$$: $$f(x) = 2x^2 + 3x - 2 + \frac{\sin 2x}{2}$$

Find critical points on [0, 1/2].

$$f'(x) = -4x - 3 + \cos 2x$$

At $$x = 0$$: $$f'(0) = -3 + 1 = -2 < 0$$. Since $$\cos 2x \leq 1$$ and $$-4x - 3 \leq -3$$, we have $$f'(x) < 0$$ on $$(0, 1/2)$$.

So $$f$$ is strictly decreasing on $$[0, 1/2]$$.

Find critical points on [1/2, 1].

$$f'(x) = 4x + 3 + \cos 2x$$

Since $$4x + 3 \geq 5$$ for $$x \geq 1/2$$ and $$\cos 2x \geq -1$$, we have $$f'(x) \geq 4 > 0$$.

So $$f$$ is strictly increasing on $$[1/2, 1]$$.

Evaluate at key points.

$$f(0) = 2 + 0 = 2$$

$$f(1/2) = 0 + \frac{\sin 1}{2} = \frac{\sin 1}{2}$$

$$f(1) = 2 + 3 - 2 + \frac{\sin 2}{2} = 3 + \frac{\sin 2}{2}$$

Determine absolute max and min.

$$f$$ decreases from $$f(0) = 2$$ to $$f(1/2) = \frac{\sin 1}{2}$$, then increases to $$f(1) = 3 + \frac{\sin 2}{2}$$.

Absolute minimum = $$\frac{\sin 1}{2}$$ (at $$x = 1/2$$).

Absolute maximum = $$3 + \frac{\sin 2}{2}$$ (at $$x = 1$$).

Compute the sum.

$$\text{Sum} = \frac{\sin 1}{2} + 3 + \frac{\sin 2}{2} = 3 + \frac{\sin 1 + \sin 2}{2}$$

Using $$\sin 2 = 2\sin 1 \cos 1$$:

$$= 3 + \frac{\sin 1 + 2\sin 1 \cos 1}{2} = 3 + \frac{\sin 1(1 + 2\cos 1)}{2}$$

Answer: Option B

The surface area of a sphere is $$S = 4\pi r^2$$. The surface area increases at a constant rate, so $$\frac{dS}{dt} = k$$ (constant).

Express S as a linear function of time.

Since $$\frac{dS}{dt} = k$$, we have $$S(t) = S(0) + kt$$.

Use the given conditions.

At $$t = 0$$: $$r = 3$$, so $$S(0) = 4\pi(3^2) = 36\pi$$.

At $$t = 5$$: $$r = 7$$, so $$S(5) = 4\pi(7^2) = 196\pi$$.

$$k = \frac{196\pi - 36\pi}{5} = \frac{160\pi}{5} = 32\pi$$

Find the radius at t = 9.

$$S(9) = 36\pi + 9 \times 32\pi = 36\pi + 288\pi = 324\pi$$

$$4\pi r^2 = 324\pi$$

$$r^2 = 81$$

$$r = 9$$

Answer: Option A (r = 9)

Given

$$\vec a=(\alpha,1,\beta)$$

and

$$\vec b=(3,-5,4)$$

Compute the cross product:

$$\vec a\times\vec b=\begin{vmatrix}\hat i&\hat j&\hat k\\ \alpha&1&\beta\\ 3&-5&4\end{vmatrix}$$

$$=(4+5\beta)\hat i-(4\alpha-3\beta)\hat j+(-5\alpha-3)\hat k$$

Comparing with

$$-\hat i+9\hat j+12\hat k$$

we get

$$4+5\beta=-1$$

$$\beta=-1$$

Also,

$$-5\alpha-3=12$$

$$\alpha=-3$$

Hence

$$\vec a=-3\hat i+\hat j-\hat k$$

Now,

$$\vec b-2\vec a=(3,-5,4)-(-6,2,-2)=(9,-7,6)$$

and

$$\vec b+\vec a=(3,-5,4)+(-3,1,-1)=(0,-4,3)$$

The scalar projection of

$$\vec b-2\vec a$$

on

$$\vec b+\vec a$$

is

$$\frac{(\vec b-2\vec a)\cdot(\vec b+\vec a)}{|\vec b+\vec a|}$$

Now,

$$ (9,-7,6)\cdot(0,-4,3)=0+28+18=46 $$

and

$$ |\vec b+\vec a|=\sqrt{0^2+(-4)^2+3^2}=5 $$

Therefore the required projection is

$$\frac{46}{5}$$

Final Answer :

$$\frac{46}{5}$$

The incorrect mean is

$$15$$

for

$$10$$

observations.

Therefore, the incorrect sum is

$$10\times 15=150$$

Since

$$25$$

was taken instead of

$$15$$,

the sum is excess by

$$10$$.

Hence the correct sum is

$$150-10=140$$

Therefore, the correct mean is

$$\bar x=\frac{140}{10}=14$$

Now use the variance formula

$$\sigma^2=\frac{\sum x_i^2}{n}-\bar x^2$$

The incorrect variance is

$$15$$

with incorrect mean

$$15$$.

Hence

$$15=\frac{\sum x_i^2}{10}-15^2$$

$$\frac{\sum x_i^2}{10}=240$$

$$\sum x_i^2=2400$$

Since

$$25$$

was used instead of

$$15$$,

the correct value of

$$\sum x_i^2$$

is

$$2400-25^2+15^2$$

$$=2400-625+225$$

$$=2000$$

Therefore, the correct variance is

$$\sigma^2=\frac{2000}{10}-14^2$$

$$=200-196$$

$$=4$$

Hence the correct standard deviation is

$$\sqrt{4}$$

$$=2$$

Final Answer :

$$2$$

Given,

$$f(x)=|5x-7|+[x^2+2x],\qquad x\in\left[\frac54,2\right]$$

First analyze the modulus term.

$$5x-7=0\iff x=\frac75$$

Hence,

Case 1: $$\frac54\le x<\frac75$$

$$|5x-7|=7-5x$$

Case 2: $$\frac75\le x\le2$$

$$|5x-7|=5x-7$$

Now analyze

$$[x^2+2x]$$

Let

$$g(x)=x^2+2x$$

Since

$$g'(x)=2x+2>0,$$

$$g(x)$$ is increasing on the interval.

Now,

$$g\left(\frac54\right)=\frac{25}{16}+\frac52=\frac{65}{16}$$

$$=4.0625$$

and

$$g(2)=8$$

Hence,

$$[x^2+2x]$$

takes values

$$4,5,6,7,8$$

Now find transition points:

$$x^2+2x=5$$

$$x^2+2x-5=0$$

$$x=-1+\sqrt6$$

Similarly,

$$x^2+2x=6$$

$$x=-1+\sqrt7$$

and

$$x^2+2x=7$$

$$x=-1+2\sqrt2$$

Thus,

$$f(x)=7-5x+4=11-5x,\quad \frac54\le x<\frac75$$

This is decreasing.

Hence maximum in this part occurs at

$$x=\frac54$$

giving

$$f\left(\frac54\right)=11-\frac{25}{4}=\frac{19}{4}$$

Now for

$$\frac75\le x<-1+\sqrt6,$$

$$f(x)=5x-7+4=5x-3$$

which is increasing.

At

$$x=\frac75,$$

$$f\left(\frac75\right)=4$$

Next intervals:

$$f(x)=5x-2,\quad -1+\sqrt6\le x<-1+\sqrt7$$

$$f(x)=5x-1,\quad -1+\sqrt7\le x<-1+2\sqrt2$$

$$f(x)=5x,\quad -1+2\sqrt2\le x<2$$

and at

$$x=2,$$

$$f(2)=|10-7|+[8]=3+8=11$$

Hence minimum value is

$$4$$

and maximum value is

$$11$$

Therefore,

$$4+11=15$$

Hence, the required sum is

$$\boxed{15}$$.

A right circular cone has its vertex downward and semivertical angle $$\theta = \tan^{-1}\frac{3}{4}$$. Water is poured into the cone at a constant rate of 6 cubic meters per hour, and we seek the rate of increase of the wet curved surface area when the depth of the water is $$h = 4$$ meters.

Denote by $$h the depth of the water and by r the radius of the water surface. Since \tan\theta = \frac{3}{4}$$ and the triangle formed by the radius and the depth is similar to the triangle defined by the semivertical angle, we have $$\frac{r}{h} = \frac{3}{4}, hence r = \frac{3h}{4}$$.

The volume of water in the cone is given by $$V = \frac{1}{3}\pi r^2 h, which on substituting for r becomes V = \frac{1}{3}\pi\left(\frac{3h}{4}\right)^2 h = \frac{3\pi h^3}{16}\,. Differentiating with respect to time t yields \frac{dV}{dt} = \frac{9\pi h^2}{16}\cdot\frac{dh}{dt}\,, and since \frac{dV}{dt} = 6, we have \frac{9\pi h^2}{16}\frac{dh}{dt} = 6\,. At h = 4 this gives 9\pi\frac{dh}{dt} = 6 and thus \frac{dh}{dt} = \frac{2}{3\pi}\,.$$

Next, the slant height $$l of the water surface satisfies l = \frac{h}{\cos\theta}. From \tan\theta = \frac{3}{4} one finds \sin\theta = \frac{3}{5} and \cos\theta = \frac{4}{5}, so l = \frac{5h}{4}\,. The wet curved surface area is S = \pi r\,l = \pi\cdot\frac{3h}{4}\cdot\frac{5h}{4} = \frac{15\pi h^2}{16}\,. $$

Differentiating $$S with respect to time gives \frac{dS}{dt} = \frac{15\pi\cdot 2h}{16}\cdot\frac{dh}{dt} = \frac{30\pi h}{16}\cdot\frac{dh}{dt}\,. Substituting h = 4 and \frac{dh}{dt} = \frac{2}{3\pi} yields \frac{dS}{dt} = \frac{30\pi\cdot 4}{16}\cdot\frac{2}{3\pi} = 5\,. $$ Therefore, the wet curved surface area is increasing at the rate $$5$$ square meters per hour.

We are given that $$f$$ and $$g$$ are twice differentiable even functions on $$(-2, 2)$$ satisfying $$f(1/4) = 0$$, $$f(1/2) = 0$$, $$f(1) = 1$$, $$g(3/4) = 0$$, and $$g(1) = 2$$, and we want to determine the minimum number of solutions of $$f(x)g''(x) + f'(x)g'(x) = 0$$ in $$(-2, 2)$$. Noting that $$f(x)g''(x) + f'(x)g'(x) = \frac{d}{dx}[f(x)g'(x)]$$, we set $$h(x) = f(x)g'(x)$$ and focus on the zeros of $$h'(x)$$.

Since $$f$$ is even, its zeros at $$x=1/4$$ and $$x=1/2$$ also occur at $$x=-1/4$$ and $$x=-1/2$$. The evenness of $$g$$ makes $$g'$$ odd and hence $$g'(0)=0$$. In addition, $$g(3/4) = 0$$ and $$g(1) = 2$$ together with Rolle’s theorem on the interval $$[-3/4,3/4]$$ confirm at least one zero of $$g'$$ in $$(-3/4,3/4)$$, which we already have at $$x=0$$.

Therefore, the zeros of $$h(x)=f(x)g'(x)$$ occur at least at $$x=-1/2,-1/4,0,1/4,1/2$$, giving five zeros. By Rolle’s theorem, between each pair of consecutive zeros of $$h$$ there is at least one zero of $$h'(x)$$, namely in each of the intervals $$(-1/2,-1/4)$$, $$(-1/4,0)$$, $$(0,1/4)$$, and $$(1/4,1/2)$$. Hence there are at least four zeros of $$h'(x)$$ in $$(-2,2)$$, and so the minimum number of solutions of the given equation is 4.

Given,

$$f(x)=x|x^2-1|-2|x-3|+x-3,\qquad x\in(0,4)$$

The modulus terms change sign at

$$x=1,\qquad x=3$$

So, consider the function case-wise.

Case 1: $$0<x<1$$

Here,

$$x^2-1<0,\qquad x-3<0$$

Thus,

$$|x^2-1|=1-x^2,\qquad |x-3|=3-x$$

Hence,

$$f(x)=x(1-x^2)-2(3-x)+x-3$$

$$=-x^3+4x-9$$

Therefore,

$$f'(x)=-3x^2+4$$

Now,

$$f'(x)=0\Rightarrow -3x^2+4=0$$

$$x=\frac{2}{\sqrt3}>1$$

So, there is no critical point in $$(0,1)$$.

Also,

$$f'(x)>0\quad \text{for all}\quad x\in(0,1)$$

Hence, $$f$$ is increasing on $$(0,1)$$.

Case 2: $$1<x<3$$

Here,

$$x^2-1>0,\qquad x-3<0$$

Thus,

$$|x^2-1|=x^2-1,\qquad |x-3|=3-x$$

Hence,

$$f(x)=x(x^2-1)-2(3-x)+x-3$$

$$=x^3+2x-9$$

Therefore,

$$f'(x)=3x^2+2>0$$

Hence, $$f$$ is increasing on $$(1,3)$$.

Case 3: $$3<x<4$$

Here,

$$x^2-1>0,\qquad x-3>0$$

Thus,

$$|x^2-1|=x^2-1,\qquad |x-3|=x-3$$

Hence,

$$f(x)=x(x^2-1)-2(x-3)+x-3$$

$$=x^3-2x+3$$

Therefore,

$$f'(x)=3x^2-2>0$$

Hence, $$f$$ is increasing on $$(3,4)$$.

Now, check the corner points.

At

$$x=1$$

$$f'_-(1)=1,\qquad f'_+(1)=5$$

Both are positive, so no local extremum occurs at $$x=1$$.

At

$$x=3$$

$$f'_-(3)=29,\qquad f'_+(3)=25$$

Again, both are positive, so no local extremum occurs at $$x=3$$.

Thus, the function is increasing throughout $$(0,4)$$.

Therefore,

$$m=0,\qquad M=0$$

Hence, $$\boxed{m+M=0}$$.

We need to find the area enclosed by the $$x$$-axis, and the tangent and normal to the curve $$4x^3 - 3xy^2 + 6x^2 - 5xy - 8y^2 + 9x + 14 = 0$$ at $$(-2, 3)$$.

Verifying that the point $$(-2, 3)$$ lies on the curve gives:

$$4(-8) - 3(-2)(9) + 6(4) - 5(-2)(3) - 8(9) + 9(-2) + 14 = -32 + 54 + 24 + 30 - 72 - 18 + 14 = 0 \checkmark$$

Implicit differentiation of the curve yields:

$$12x^2 - 3y^2 - 6xyy' + 12x - 5y - 5xy' - 16yy' + 9 = 0$$

$$y'(-6xy - 5x - 16y) = -12x^2 + 3y^2 - 12x + 5y - 9$$

Substituting $$(-2, 3)$$ into this gives

$$y'(36 + 10 - 48) = -48 + 27 + 24 + 15 - 9$$

$$y'(-2) = 9 \implies y' = -\frac{9}{2}$$

The tangent line at $$(-2, 3)$$ has slope $$-\frac{9}{2}$$ and is given by

$$y - 3 = -\frac{9}{2}(x + 2) \implies 9x + 2y + 12 = 0$$

Its $$x$$-intercept (set $$y = 0$$) is $$x = -\frac{4}{3}$$, yielding the point $$\left(-\frac{4}{3}, 0\right)$$.

The normal line at $$(-2, 3)$$ has slope $$\frac{2}{9}$$ and is given by

$$y - 3 = \frac{2}{9}(x + 2) \implies 2x - 9y + 31 = 0$$

Its $$x$$-intercept (set $$y = 0$$) is $$x = -\frac{31}{2}$$, yielding the point $$\left(-\frac{31}{2}, 0\right)$$.

The triangle formed by the points $$(-2, 3)$$, $$\left(-\frac{4}{3}, 0\right)$$, and $$\left(-\frac{31}{2}, 0\right)$$ has a base along the $$x$$-axis of length $$\left|-\frac{4}{3} + \frac{31}{2}\right| = \left|\frac{-8 + 93}{6}\right| = \frac{85}{6}$$ and a height of $$3$$.

Thus $$A = \frac{1}{2} \times \frac{85}{6} \times 3 = \frac{85}{4}$$, and hence $$8A = 8 \times \frac{85}{4} = 170$$.

The answer is $$\boxed{170}$$.

We have the curve $$y = x^3 - x^2 + x$$ and we need the tangent at $$(a, b)$$ on this curve that is also tangent to $$y = 5x^2 + 2x - 25$$ at the point $$(2, -1)$$.

First, we verify $$(2, -1)$$ lies on the second curve: $$5(4) + 2(2) - 25 = 20 + 4 - 25 = -1$$. Yes.

The slope of the tangent to $$y = 5x^2 + 2x - 25$$ at $$x = 2$$ is $$y' = 10x + 2 = 22$$. So the tangent line at $$(2, -1)$$ is $$y - (-1) = 22(x - 2)$$, giving $$y = 22x - 45$$.

This same line is tangent to $$y = x^3 - x^2 + x$$ at $$(a, b)$$. Since $$(a,b)$$ is on the first curve, $$b = a^3 - a^2 + a$$. The slope at $$x = a$$ is $$3a^2 - 2a + 1 = 22$$, so $$3a^2 - 2a - 21 = 0$$. Using the quadratic formula: $$a = \frac{2 \pm \sqrt{4 + 252}}{6} = \frac{2 \pm 16}{6}$$, giving $$a = 3$$ or $$a = -\frac{7}{3}$$.

We check which value gives a tangent line passing through $$(2, -1)$$. The tangent at $$(a, b)$$ is $$y - b = 22(x - a)$$.

Case 1: $$a = 3$$: $$b = 27 - 9 + 3 = 21$$. Tangent: $$y - 21 = 22(x - 3)$$, so $$y = 22x - 45$$. At $$x = 2$$: $$y = 44 - 45 = -1$$. This matches.

Case 2: $$a = -\frac{7}{3}$$: $$b = -\frac{343}{27} - \frac{49}{9} - \frac{7}{3} = -\frac{343}{27} - \frac{147}{27} - \frac{63}{27} = -\frac{553}{27}$$. Tangent: $$y + \frac{553}{27} = 22\!\left(x + \frac{7}{3}\right)$$. At $$x = 2$$: $$y = 22 \cdot 2 + \frac{154}{3} - \frac{553}{27} = 44 + \frac{1386}{27} - \frac{553}{27} = 44 + \frac{833}{27} \neq -1$$.

So $$a = 3, b = 21$$. Therefore $$|2a + 9b| = |6 + 189| = 195$$.

Hence, the correct answer is $$\boxed{195}$$.

We have $$f: [0,1] \to \mathbb{R}$$, twice differentiable in $$(0,1)$$, with $$f(0) = 3$$ and $$f(1) = 5$$. The line $$y = 2x + 3$$ intersects the graph of $$f$$ at exactly two distinct points in $$(0,1)$$.

At $$x = 0$$: $$y = 3 = f(0)$$. At $$x = 1$$: $$y = 5 = f(1)$$.

So the line connects the endpoints of $$f$$.

Then $$h(0) = 0$$ and $$h(1) = 0$$.

The line intersects $$f$$ at exactly two points in $$(0,1)$$, so $$h(x) = 0$$ at exactly two points in $$(0,1)$$. Combined with the endpoints, $$h$$ has at least 4 zeros in $$[0,1]$$: $$x = 0$$, two interior points, and $$x = 1$$.

With 4 zeros of $$h$$, by Rolle's theorem, $$h'(x) = 0$$ at least 3 times in $$(0,1)$$.

Applying Rolle's theorem again to $$h'$$, we get $$h''(x) = 0$$ at least 2 times in $$(0,1)$$.

Since $$h''(x) = f''(x) - 0 = f''(x)$$, we conclude that $$f''(x) = 0$$ at least 2 times.

The least number of points where $$f''(x) = 0$$ is 2.

The answer is 2.

The curve is $$y = 2x^2 + x + 2$$. At point $$P(a, 2a^2 + a + 2)$$, the slope of the tangent is:

$$\frac{dy}{dx} = 4x + 1 \implies \text{slope at } P = 4a + 1$$

The normal at $$P$$ has slope $$= -\frac{1}{4a+1}$$.

Since $$Q(6, 4)$$ lies on the normal:

$$\frac{4 - (2a^2 + a + 2)}{6 - a} = -\frac{1}{4a+1}$$

$$\frac{2 - 2a^2 - a}{6 - a} = -\frac{1}{4a+1}$$

Cross-multiplying:

$$(2 - 2a^2 - a)(4a + 1) = -(6 - a)$$

$$(2 - 2a^2 - a)(4a+1) = a - 6$$

Expanding the left side:

$$8a + 2 - 8a^3 - 2a^2 - 4a^2 - a = a - 6$$

$$-8a^3 - 6a^2 + 7a + 2 = a - 6$$

$$-8a^3 - 6a^2 + 6a + 8 = 0$$

$$8a^3 + 6a^2 - 6a - 8 = 0$$

$$4a^3 + 3a^2 - 3a - 4 = 0$$

Testing $$a = 1$$: $$4 + 3 - 3 - 4 = 0$$. So $$a = 1$$ is a root.

Factoring: $$4a^3 + 3a^2 - 3a - 4 = (a - 1)(4a^2 + 7a + 4)$$

The discriminant of $$4a^2 + 7a + 4$$ is $$49 - 64 = -15 < 0$$, so no other real roots.

Thus $$P = (1, 5)$$.

Area of triangle $$OPQ$$ with $$O(0,0)$$, $$P(1, 5)$$, $$Q(6, 4)$$:

$$\text{Area} = \frac{1}{2}|x_O(y_P - y_Q) + x_P(y_Q - y_O) + x_Q(y_O - y_P)|$$

$$= \frac{1}{2}|0(5-4) + 1(4-0) + 6(0-5)|$$

$$= \frac{1}{2}|0 + 4 - 30| = \frac{1}{2} \times 26 = 13$$

Hence the answer is $$\boxed{13}$$.

Let $$t = a^x$$, where $$t > 0$$ since $$a > 0$$. Then $$f(x) = a^t + a^{1 - t} = a^t + \frac{a}{a^t}$$.

Applying the AM-GM inequality to the two positive quantities $$a^t$$ and $$\frac{a}{a^t}$$, we get $$a^t + \frac{a}{a^t} \geq 2\sqrt{a^t \cdot \frac{a}{a^t}} = 2\sqrt{a}$$.

Equality holds when $$a^t = \frac{a}{a^t}$$, i.e., $$a^{2t} = a$$, which gives $$t = \frac{1}{2}$$. Since $$t = a^x$$, this requires $$a^x = \frac{1}{2}$$, which is achievable for a suitable real $$x$$.

Therefore, the minimum value of $$f(x)$$ is $$2\sqrt{a}$$.

We need to find the set $$A = \{x \in R : f \text{ is increasing}\}$$ for the piecewise function.

For $$x < -5$$: $$f(x) = -55x$$, so $$f'(x) = -55 < 0$$. The function is decreasing in this interval.

For $$-5 \leq x \leq 4$$: $$f(x) = 2x^3 - 3x^2 - 120x$$, so $$f'(x) = 6x^2 - 6x - 120 = 6(x^2 - x - 20) = 6(x - 5)(x + 4)$$. This is positive when $$x < -4$$ or $$x > 5$$. Within $$[-5, 4]$$, the derivative is positive for $$-5 \leq x < -4$$ and negative for $$-4 < x \leq 4$$.

For $$x > 4$$: $$f(x) = 2x^3 - 3x^2 - 36x - 336$$, so $$f'(x) = 6x^2 - 6x - 36 = 6(x^2 - x - 6) = 6(x - 3)(x + 2)$$. This is positive when $$x > 3$$ or $$x < -2$$. Since $$x > 4$$, the derivative is positive for all $$x > 4$$.

We verify continuity at $$x = -5$$: from the left, $$f(-5) = -55(-5) = 275$$; from the middle piece, $$f(-5) = 2(-125) - 3(25) - 120(-5) = -250 - 75 + 600 = 275$$. At $$x = 4$$: from the middle piece, $$f(4) = 2(64) - 3(16) - 120(4) = 128 - 48 - 480 = -400$$; from the right piece, $$f(4) = 128 - 48 - 144 - 336 = -400$$. Both match, confirming continuity.

Combining, $$f$$ is increasing on $$(-5, -4)$$ and $$(4, \infty)$$.

Therefore, $$A = (-5, -4) \cup (4, \infty)$$.

Let us begin by describing the geometry of the situation. From a rectangle of length $$a$$ and breadth $$b$$, we remove four identical squares, each of side $$x$$, one from every corner. After removing these squares we fold the resulting flaps upward to obtain an open-top box. In this box

length $$=a-2x$$,   breadth $$=b-2x$$,   height $$=x$$.

Therefore the volume $$V$$ of the box, expressed as a function of $$x$$, is

$$V(x)=x\,(a-2x)\,(b-2x).$$

We expand the product so that differentiation becomes simple:

$$\begin{aligned} V(x)&=x\bigl[(a-2x)(b-2x)\bigr] \\ &=x\bigl[ab-2ax-2bx+4x^2\bigr] \\ &=abx-2a x^{2}-2b x^{2}+4x^{3} \\ &=abx-2(a+b)x^{2}+4x^{3}. \end{aligned}$$

To find the value of $$x$$ that gives the maximum volume, we differentiate $$V$$ with respect to $$x$$ and set the derivative to zero. The formula we use is: “The maximum (or minimum) of a differentiable function occurs where its derivative is zero.”

First derivative:

$$\begin{aligned} \frac{dV}{dx}&=\frac{d}{dx}\bigl[abx-2(a+b)x^{2}+4x^{3}\bigr] \\ &=ab-4(a+b)x+12x^{2}. \end{aligned}$$

Setting $$\dfrac{dV}{dx}=0$$ for extreme values gives

$$ab-4(a+b)x+12x^{2}=0.$$

We rewrite this quadratic in the standard form $$Ax^{2}+Bx+C=0$$:

$$12x^{2}-4(a+b)x+ab=0.$$

Dividing every term by $$4$$ to simplify, we obtain

$$3x^{2}-(a+b)x+\frac{ab}{4}=0.$$

Now we apply the quadratic‐formula statement: “For $$\alpha x^{2}+\beta x+\gamma=0$$, the roots are $$x=\dfrac{-\beta\pm\sqrt{\beta^{2}-4\alpha\gamma}}{2\alpha}.$$” Here $$\alpha=3$$, $$\beta=-(a+b)$$ and $$\gamma=\dfrac{ab}{4}$$. Substituting, we get

$$\begin{aligned} x&=\frac{(a+b)\pm\sqrt{(a+b)^{2}-3ab}}{6}. \end{aligned}$$

We simplify the discriminant inside the square root:

$$\begin{aligned} (a+b)^{2}-3ab&=a^{2}+2ab+b^{2}-3ab\\ &=a^{2}+b^{2}-ab. \end{aligned}$$

Hence the critical values of $$x$$ are

$$x=\frac{\,a+b\pm\sqrt{a^{2}+b^{2}-ab}\,}{6}.$$

Next, we decide which of the two roots is admissible. Because the squares we cut out must actually fit inside the original rectangle, we need $$0<x<\dfrac{a}{2}$$ and $$0<x<\dfrac{b}{2}$$. The expression with the plus sign, $$\dfrac{a+b+\sqrt{a^{2}+b^{2}-ab}}{6}$$, is evidently larger than $$\dfrac{a+b}{6}$$ and in many practical configurations exceeds either $$\dfrac{a}{2}$$ or $$\dfrac{b}{2}$$, violating the geometric constraint. The expression with the minus sign, however, is always the smaller positive root and therefore satisfies the condition of fitting inside both $$\dfrac{a}{2}$$ and $$\dfrac{b}{2}.$$

Thus the only feasible choice is

$$x=\frac{a+b-\sqrt{a^{2}+b^{2}-ab}}{6}.$$

We still have to confirm that this root corresponds to a maximum volume. For that we examine the second derivative:

$$\frac{d^{2}V}{dx^{2}}=\frac{d}{dx}\bigl[ab-4(a+b)x+12x^{2}\bigr]=-4(a+b)+24x.$$

Substituting our admissible value of $$x$$, which is smaller than $$\dfrac{a+b}{6}$$, makes $$24x<4(a+b)$$, so

$$\frac{d^{2}V}{dx^{2}}=-4(a+b)+24x<0.$$

A negative second derivative at the critical point tells us that the volume indeed attains a maximum there.

Therefore, the value of $$x$$ that maximizes the volume of the open-top box is

$$x=\frac{a+b-\sqrt{a^{2}+b^{2}-ab}}{6}.$$

Hence, the correct answer is Option C.

We have $$f(x) = 3\ln\left|\frac{x-1}{x+1}\right| - \frac{2}{x-1}$$, defined on $$\mathbb{R} \setminus \{-1, 1\}$$.

Differentiating: $$f'(x) = 3\left(\frac{1}{x-1} - \frac{1}{x+1}\right) + \frac{2}{(x-1)^2} = \frac{3}{x-1} - \frac{3}{x+1} + \frac{2}{(x-1)^2}$$.

Taking the common denominator $$(x-1)^2(x+1)$$:

$$f'(x) = \frac{3(x-1)(x+1) - 3(x-1)^2 + 2(x+1)}{(x-1)^2(x+1)}$$

Expanding the numerator: $$3(x^2 - 1) - 3(x^2 - 2x + 1) + 2(x + 1) = 3x^2 - 3 - 3x^2 + 6x - 3 + 2x + 2 = 8x - 4 = 4(2x - 1)$$.

So $$f'(x) = \frac{4(2x-1)}{(x-1)^2(x+1)}$$. Since $$(x-1)^2 > 0$$ for $$x \neq 1$$, the sign of $$f'(x)$$ depends on $$\frac{2x-1}{x+1}$$.

$$f'(x) \geq 0$$ when $$\frac{2x-1}{x+1} \geq 0$$, which happens when both factors are non-negative or both non-positive:

Case 1: $$2x - 1 \geq 0$$ and $$x + 1 > 0$$, i.e., $$x \geq \frac{1}{2}$$ (with $$x \neq 1$$).

Case 2: $$2x - 1 \leq 0$$ and $$x + 1 < 0$$, i.e., $$x < -1$$.

Therefore, $$f(x)$$ is increasing on $$(-\infty, -1) \cup \left[\frac{1}{2}, \infty\right) \setminus \{1\}$$.

We have $$f(x) = \frac{4x^3 - 3x^2}{6} - 2\sin x + (2x - 1)\cos x$$.

Differentiating term by term: $$\frac{d}{dx}\left(\frac{4x^3 - 3x^2}{6}\right) = \frac{12x^2 - 6x}{6} = 2x^2 - x$$.

$$\frac{d}{dx}(-2\sin x) = -2\cos x$$.

$$\frac{d}{dx}((2x - 1)\cos x) = 2\cos x - (2x - 1)\sin x$$.

So $$f'(x) = 2x^2 - x - 2\cos x + 2\cos x - (2x - 1)\sin x = 2x^2 - x - (2x - 1)\sin x$$.

Factoring: $$f'(x) = x(2x - 1) - (2x - 1)\sin x = (2x - 1)(x - \sin x)$$.

We know that $$x - \sin x \geq 0$$ for $$x \geq 0$$ and $$x - \sin x \leq 0$$ for $$x \leq 0$$ (since the derivative of $$x - \sin x$$ is $$1 - \cos x \geq 0$$, so $$x - \sin x$$ is increasing with value 0 at $$x = 0$$).

For $$x > \frac{1}{2}$$: $$(2x - 1) > 0$$ and $$(x - \sin x) > 0$$, so $$f'(x) > 0$$. The function is increasing on $$\left(\frac{1}{2}, \infty\right)$$.

Hence, the correct answer is Option A.

We are given the function $$f(x) = x^3 - 6x^2 + ax + b$$ such that $$f(2) = f(4) = 0$$.

Substituting $$x = 2$$ into the function: $$f(2) = 2^3 - 6(2^2) + a(2) + b = 8 - 24 + 2a + b = 2a + b - 16 = 0$$, which gives the equation $$2a + b = 16$$.

Substituting $$x = 4$$ into the function: $$f(4) = 4^3 - 6(4^2) + a(4) + b = 64 - 96 + 4a + b = 4a + b - 32 = 0$$, which gives the equation $$4a + b = 32$$.

Solving these two linear equations simultaneously, we subtract the first from the second: $$(4a + b) - (2a + b) = 32 - 16$$, which results in $$2a = 16$$ or $$a = 8$$.

Substituting $$a = 8$$ into $$2a + b = 16$$ gives $$16 + b = 16$$, so $$b = 0$$.

Thus, the function is $$f(x) = x^3 - 6x^2 + 8x$$.

Differentiating $$f(x)$$ with respect to $$x$$, we get the derivative function: $$f'(x) = 3x^2 - 12x + 8$$.

Now let us analyze statement $$S_1$$: "there exists $$x_1, x_2 \in (2, 4), x_1 < x_2$$, such that $$f'(x_1) = -1$$ and $$f'(x_2) = 0$$."

At the endpoints, $$f(2) = 0$$ and $$f(4) = 0$$. By Rolle's Theorem, since $$f(x)$$ is continuous on $$[2, 4]$$ and differentiable on $$(2, 4)$$, there exists a value $$c \in (2, 4)$$ such that $$f'(c) = 0$$.

Solving $$3x^2 - 12x + 8 = 0$$, we find the roots using the quadratic formula: $$x = \frac{12 \pm \sqrt{144 - 96}}{6} = \frac{12 \pm 4\sqrt{3}}{6} = 2 \pm \frac{2}{\sqrt{3}}$$.

Taking $$x_2 = 2 + \frac{2}{\sqrt{3}} \approx 3.15$$, this value lies in the interval $$(2, 4)$$ and satisfies $$f'(x_2) = 0$$.

To check for $$f'(x_1) = -1$$, we evaluate the derivative at the lower endpoint: $$f'(2) = 3(4) - 12(2) + 8 = 12 - 24 + 8 = -4$$.

Since $$f'(2) = -4$$ and $$f'(x_2) = 0$$, by the Intermediate Value Theorem for continuous derivatives, there exists $$x_1 \in (2, x_2)$$ such that $$f'(x_1) = -1$$. Therefore, statement $$S_1$$ is true.

Now let us analyze statement $$S_2$$: "there exists $$x_3, x_4 \in (2, 4), x_3 < x_4$$, such that $$f$$ is decreasing in $$(2, x_4)$$, increasing in $$(x_4, 4)$$ and $$2f'(x_3) = \sqrt{3}f(x_4)$$."

For monotonicity, we examine the sign of $$f'(x) = 3x^2 - 12x + 8$$. The critical point in $$(2, 4)$$ is $$x_4 = 2 + \frac{2}{\sqrt{3}}$$. Since $$f'(2) = -4 < 0$$ and the parabola opens upwards, $$f(x)$$ is decreasing on $$(2, x_4)$$ and increasing on $$(x_4, 4)$$.

Let us compute $$f(x_4)$$ at $$x_4 = 2 + \frac{2}{\sqrt{3}}$$: $$f(x_4) = x_4(x_4 - 2)(x_4 - 4) = \left(2 + \frac{2}{\sqrt{3}}\right) \left(\frac{2}{\sqrt{3}}\right) \left(\frac{2}{\sqrt{3}} - 2\right) = \frac{-16}{3\sqrt{3}}$$.

Thus, the right side of the condition is $$\sqrt{3}f(x_4) = \sqrt{3}\left(-\frac{16}{3\sqrt{3}}\right) = -\frac{16}{3}$$.

We need $$2f'(x_3) = -\frac{16}{3}$$, which means $$f'(x_3) = -\frac{8}{3} \approx -2.67$$.

Since $$f'(2) = -4$$ and $$f'(x_4) = 0$$, and $$-\frac{8}{3}$$ lies between $$-4$$ and $$0$$, by the Intermediate Value Theorem, there exists $$x_3 \in (2, x_4)$$ such that $$f'(x_3) = -\frac{8}{3}$$.

Since $$x_3 \in (2, x_4)$$ and $$x_4 < 4$$, we have $$x_3 < x_4$$. Therefore, statement $$S_2$$ is true.

Hence both statements $$S_1$$ and $$S_2$$ are true, making the correct option C.

Given $$f(x) = x^3 - 3x^2 - \frac{3f''(2)}{2}x + f''(1)$$, we first determine $$f''(2)$$ and $$f''(1)$$.

Differentiating twice: $$f'(x) = 3x^2 - 6x - \frac{3f''(2)}{2}$$ and $$f''(x) = 6x - 6$$.

Evaluating: $$f''(2) = 12 - 6 = 6$$ and $$f''(1) = 6 - 6 = 0$$.

Substituting back: $$f(x) = x^3 - 3x^2 - \frac{3 \cdot 6}{2}x + 0 = x^3 - 3x^2 - 9x.$$

Critical points: $$f'(x) = 3x^2 - 6x - 9 = 3(x-3)(x+1) = 0$$, giving $$x = 3$$ and $$x = -1$$.

Using the second derivative test: $$f''(3) = 12 > 0$$ (local minimum) and $$f''(-1) = -12 < 0$$ (local maximum).

The only local minimum value is $$f(3) = 27 - 27 - 27 = -27$$.

The sum of all local minimum values is $$-27$$.

We need to find the triangle of maximum area that can be inscribed in a circle of radius $$r$$. Let the circle be centred at the origin with equation $$x^2 + y^2 = r^2$$. We parameterize the three vertices as $$A = (r\cos\theta_1, r\sin\theta_1)$$, $$B = (r\cos\theta_2, r\sin\theta_2)$$, and $$C = (r\cos\theta_3, r\sin\theta_3)$$.

Without loss of generality, fix one vertex at $$A = (r, 0)$$ (i.e., $$\theta_1 = 0$$), and let $$\theta_2 = \alpha$$ and $$\theta_3 = \alpha + \beta$$, where $$\alpha > 0$$, $$\beta > 0$$, and $$\alpha + \beta < 2\pi$$. The area of the triangle using the cross-product formula is:

$$\text{Area} = \dfrac{r^2}{2}\big|\sin(\theta_2 - \theta_1) + \sin(\theta_3 - \theta_2) + \sin(\theta_1 - \theta_3)\big| = \dfrac{r^2}{2}\big(\sin\alpha + \sin\beta + \sin(2\pi - \alpha - \beta)\big)$$

Since $$\sin(2\pi - \alpha - \beta) = -\sin(\alpha + \beta)$$, we get $$\text{Area} = \dfrac{r^2}{2}\big(\sin\alpha + \sin\beta - \sin(\alpha + \beta)\big)$$.

To maximize, we take partial derivatives. Setting $$\dfrac{\partial(\text{Area})}{\partial\alpha} = 0$$: $$\cos\alpha - \cos(\alpha + \beta) = 0$$, so $$\cos\alpha = \cos(\alpha + \beta)$$. Setting $$\dfrac{\partial(\text{Area})}{\partial\beta} = 0$$: $$\cos\beta - \cos(\alpha + \beta) = 0$$, so $$\cos\beta = \cos(\alpha + \beta)$$.

From these, $$\cos\alpha = \cos\beta$$, which gives $$\alpha = \beta$$ (since both are in $$(0, 2\pi)$$). Substituting into $$\cos\alpha = \cos 2\alpha$$: $$\cos\alpha = 2\cos^2\alpha - 1$$, i.e., $$2\cos^2\alpha - \cos\alpha - 1 = 0$$. Factoring: $$(2\cos\alpha + 1)(\cos\alpha - 1) = 0$$. Since $$\alpha \neq 0$$, we have $$\cos\alpha = -\dfrac{1}{2}$$, giving $$\alpha = \dfrac{2\pi}{3}$$. The third angle subtended is $$2\pi - 2 \cdot \dfrac{2\pi}{3} = \dfrac{2\pi}{3}$$, confirming the triangle is equilateral.

The maximum area is $$\dfrac{r^2}{2}\left(3\sin\dfrac{2\pi}{3}\right) = \dfrac{r^2}{2} \cdot \dfrac{3\sqrt{3}}{2} = \dfrac{3\sqrt{3}}{4}r^2$$.

Now we find the side length. Each side is a chord subtending an angle of $$\dfrac{2\pi}{3}$$ at the centre. Using the chord formula $$\ell = 2r\sin\dfrac{\theta}{2}$$: $$s = 2r\sin\dfrac{\pi}{3} = 2r \cdot \dfrac{\sqrt{3}}{2} = \sqrt{3}\,r$$.

Checking option 3: the height of this equilateral triangle is $$h = \dfrac{\sqrt{3}}{2} \times \sqrt{3}\,r = \dfrac{3r}{2}$$, which does not equal $$\dfrac{2r}{3}$$. Checking option 2: an isosceles triangle with base $$2r$$ inscribed in the circle has its base as a diameter, making it a right triangle with area $$\dfrac{1}{2} \times 2r \times r = r^2$$, which is less than $$\dfrac{3\sqrt{3}}{4}r^2 \approx 1.30\,r^2$$. Checking option 4: a right triangle with sides $$2r$$ and $$r$$ has hypotenuse $$\sqrt{5}\,r$$, and its circumradius would be $$\dfrac{\sqrt{5}\,r}{2} \neq r$$ in general.

Therefore, the triangle of maximum area inscribed in a circle of radius $$r$$ is an equilateral triangle having each of its sides of length $$\sqrt{3}\,r$$.

Let the wire be cut in such a way that the regular hexagon obtained has side $$x\ \text{m}$$ and the square obtained has side $$y\ \text{m}$$.

The whole wire is used, so the sum of the perimeters equals the original length.

For a regular hexagon the perimeter is $$6x$$, and for a square the perimeter is $$4y$$. Hence

$$6x+4y = 20.$$

Solving for $$y$$ we get

$$4y = 20-6x$$

$$y = \frac{20-6x}{4}$$

$$y = 5 - 1.5x.$$

Next we write the expressions for the areas.

The area of a square is $$y^{2}$$.

The area of a regular hexagon of side $$x$$ is given by the standard formula

$$\text{Area}_{\text{hexagon}} = \frac{3\sqrt{3}}{2}\,x^{2}.$$

So the total area $$A(x)$$, as a function of $$x$$ alone, is

$$A(x) = y^{2} + \frac{3\sqrt{3}}{2}\,x^{2}.$$

Substituting $$y = 5-1.5x$$ we have

$$A(x) = (5-1.5x)^{2} + \frac{3\sqrt{3}}{2}\,x^{2}.$$

To minimise this area, we differentiate with respect to $$x$$ and set the derivative equal to zero.

First expand the square term:

$$(5-1.5x)^{2} = 25 - 2(5)(1.5)x + (1.5)^{2}x^{2}$$

$$ = 25 - 15x + 2.25x^{2}.$$

Therefore

$$A(x) = 25 - 15x + 2.25x^{2} + \frac{3\sqrt{3}}{2}x^{2}.$$

Combining the quadratic terms:

$$A(x) = 25 - 15x + \left(2.25 + \frac{3\sqrt{3}}{2}\right)x^{2}.$$

Differentiate:

$$\frac{dA}{dx} = -15 + 2\left(2.25 + \frac{3\sqrt{3}}{2}\right)x.$$

Simplify the coefficient of $$x$$ inside the derivative:

$$2\left(2.25 + \frac{3\sqrt{3}}{2}\right) = 4.5 + 3\sqrt{3}.$$

So

$$\frac{dA}{dx} = -15 + (4.5 + 3\sqrt{3})x.$$

Set $$\dfrac{dA}{dx}=0$$ for the extremum:

$$-15 + (4.5 + 3\sqrt{3})x = 0.$$

Hence

$$(4.5 + 3\sqrt{3})x = 15$$

$$x = \frac{15}{4.5 + 3\sqrt{3}}.$$

Write $$4.5$$ as $$\dfrac{9}{2}$$ to combine denominators:

$$4.5 + 3\sqrt{3} = \frac{9}{2} + 3\sqrt{3} = \frac{9 + 6\sqrt{3}}{2}.$$

Therefore

$$x = \frac{15}{\dfrac{9 + 6\sqrt{3}}{2}} = 15 \cdot \frac{2}{9 + 6\sqrt{3}} = \frac{30}{9 + 6\sqrt{3}}.$$

Factor out a $$3$$ from the denominator:

$$x = \frac{30}{3(3 + 2\sqrt{3})} = \frac{10}{3 + 2\sqrt{3}}.$$

Since the second derivative is positive (the coefficient of $$x^{2}$$ in $$A(x)$$ is positive), this value of $$x$$ indeed gives the minimum combined area.

Thus the side of the hexagon that minimises the total area is

$$x = \frac{10}{3 + 2\sqrt{3}}\ \text{metres}.$$

Hence, the correct answer is Option C.

We have to find the acute angle at which the ellipse $$\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$$ and the circle $$x^{2}+y^{2}=ab$$ intersect, the given condition being $$a\gt b.$$

First we locate the points of intersection. Any common point must satisfy both equations, so from the second curve we write

$$x^{2}+y^{2}=ab\;.$$

We substitute $$x^{2}=ab-y^{2}$$ in the ellipse equation:

$$\frac{ab-y^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1.$$

Multiplying every term by $$a^{2}b^{2}$$ to clear the denominators,

$$b^{2}(ab-y^{2})+a^{2}y^{2}=a^{2}b^{2}.$$

Expanding the left side,

$$ab\,b^{2}-b^{2}y^{2}+a^{2}y^{2}=a^{2}b^{2}.$$

Collecting all terms on one side,

$$ab\,b^{2}-a^{2}b^{2}+y^{2}(a^{2}-b^{2})=0.$$

Factorising the constant part,

$$-a b^{2}(a-b)+y^{2}(a^{2}-b^{2})=0.$$

The factor $$a^{2}-b^{2}=(a-b)(a+b)$$ is positive because $$a\gt b,$$ so we solve for $$y^{2}:$$

$$y^{2}=\frac{a b^{2}(a-b)}{(a-b)(a+b)}=\frac{a b^{2}}{a+b}.$$

Using $$x^{2}=ab-y^{2}$$ we get

$$x^{2}=ab-\frac{a b^{2}}{a+b}=ab\left(1-\frac{b}{a+b}\right)=ab\left(\frac{a}{a+b}\right)=\frac{a^{2}b}{a+b}.$$

Thus for every intersection point

$$x^{2}=\frac{a^{2}b}{a+b},\qquad y^{2}=\frac{a b^{2}}{a+b}.$$

Now we compute the slopes of the tangents to both curves at a common point.

Ellipse. Starting from $$F(x,y)=\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}-1=0,$$ differentiate implicitly:

$$\frac{2x}{a^{2}}+\frac{2y}{b^{2}}\frac{dy}{dx}=0.$$

Hence

$$\frac{dy}{dx}=-\frac{b^{2}x}{a^{2}y},$$

so the slope of the tangent to the ellipse is

$$m_{1}=-\frac{b^{2}x}{a^{2}y}.$$

Circle. From $$G(x,y)=x^{2}+y^{2}-ab=0$$ we differentiate:

$$2x+2y\frac{dy}{dx}=0\;\Longrightarrow\;\frac{dy}{dx}=-\frac{x}{y},$$

giving the circle’s tangent slope

$$m_{2}=-\frac{x}{y}.$$

The angle $$\theta$$ between two curves is the angle between their tangents, for which the standard relation is

$$\tan\theta=\left|\frac{m_{2}-m_{1}}{1+m_{1}m_{2}}\right|.$$

We now evaluate each part at the intersection point.

First compute $$\dfrac{x}{y}.$$ From the already-found squares,

$$\left(\frac{x}{y}\right)^{2}=\frac{x^{2}}{y^{2}}=\frac{\dfrac{a^{2}b}{\,a+b\,}}{\dfrac{a b^{2}}{\,a+b\,}}=\frac{a^{2}b}{a b^{2}}=\frac{a}{b},$$

so

$$\Bigl|\frac{x}{y}\Bigr|=\sqrt{\frac{a}{b}}.$$

Next find $$m_{1}m_{2}:$$

$$m_{1}m_{2}=\left(-\frac{b^{2}x}{a^{2}y}\right)\left(-\frac{x}{y}\right)=\frac{b^{2}x^{2}}{a^{2}y^{2}}=\frac{b^{2}}{a^{2}}\cdot\frac{x^{2}}{y^{2}}=\frac{b^{2}}{a^{2}}\cdot\frac{a}{b}=\frac{b}{a}.$$

Therefore

$$1+m_{1}m_{2}=1+\frac{b}{a}=\frac{a+b}{a}.$$

Now evaluate $$m_{2}-m_{1}:$$

$$$ \begin{aligned} m_{2}-m_{1}&=-\frac{x}{y}-\left(-\frac{b^{2}x}{a^{2}y}\right) =-\frac{x}{y}+\frac{b^{2}x}{a^{2}y} =\frac{x}{y}\left(-1+\frac{b^{2}}{a^{2}}\right)\\ &=\frac{x}{y}\left(\frac{b^{2}-a^{2}}{a^{2}}\right) =-\frac{x}{y}\cdot\frac{a^{2}-b^{2}}{a^{2}}. \end{aligned} $$$

Taking absolute value removes the minus sign, so

$$|m_{2}-m_{1}|=\Bigl|\frac{x}{y}\Bigr|\frac{a^{2}-b^{2}}{a^{2}} =\sqrt{\frac{a}{b}}\;\frac{a^{2}-b^{2}}{a^{2}}.$$

Substituting everything into the formula for $$\tan\theta$$ gives

$$$ \begin{aligned} \tan\theta &=\frac{|m_{2}-m_{1}|}{1+m_{1}m_{2}} =\frac{\sqrt{\dfrac{a}{b}}\,\dfrac{a^{2}-b^{2}}{a^{2}}}{\dfrac{a+b}{a}} =\sqrt{\frac{a}{b}}\;\frac{a^{2}-b^{2}}{a^{2}}\;\frac{a}{a+b}. \end{aligned} $$$

Simplify the factors step by step. Notice $$a^{2}-b^{2}=(a-b)(a+b),$$ so

$$$ \begin{aligned} \tan\theta &=\sqrt{\frac{a}{b}}\;\frac{(a-b)(a+b)}{a^{2}}\;\frac{a}{a+b} =\sqrt{\frac{a}{b}}\;\frac{a-b}{a}. \end{aligned} $$$

Because $$\sqrt{\dfrac{a}{b}}\cdot\dfrac{1}{a} =\frac{\sqrt{a}}{\sqrt{b}}\cdot\frac{1}{a} =\frac{1}{\sqrt{ab}},$$ we finally reach

$$\;\tan\theta=\dfrac{a-b}{\sqrt{ab}}\;.$$

Therefore the required angle of intersection is $$\theta=\tan^{-1}\dfrac{a-b}{\sqrt{ab}}$$.

Hence, the correct answer is Option C.

We have $$f(x) = \begin{cases} \left(2 - \sin\left(\frac{1}{x}\right)\right)|x|, & x \neq 0 \\ 0, & x = 0 \end{cases}$$.

For $$x > 0$$, $$|x| = x$$, so $$f(x) = \left(2 - \sin\left(\frac{1}{x}\right)\right)x = 2x - x\sin\left(\frac{1}{x}\right)$$.

Differentiating using the product rule: $$f'(x) = 2 - \sin\left(\frac{1}{x}\right) - x \cdot \cos\left(\frac{1}{x}\right) \cdot \left(-\frac{1}{x^2}\right) = 2 - \sin\left(\frac{1}{x}\right) + \frac{1}{x}\cos\left(\frac{1}{x}\right)$$.

For $$f$$ to be monotonically increasing on $$(0, \infty)$$, we need $$f'(x) \geq 0$$ for all $$x > 0$$. The term $$2 - \sin\left(\frac{1}{x}\right)$$ is always at least $$1$$ (since $$-1 \leq \sin \leq 1$$). However, the term $$\frac{1}{x}\cos\left(\frac{1}{x}\right)$$ oscillates and becomes unbounded near $$x = 0$$.

To see this, consider $$x_n = \frac{1}{(2n+1)\pi}$$ for large positive integers $$n$$. At these points, $$\frac{1}{x_n} = (2n+1)\pi$$, so $$\cos\left(\frac{1}{x_n}\right) = \cos((2n+1)\pi) = -1$$ and $$\sin\left(\frac{1}{x_n}\right) = 0$$. Then $$f'(x_n) = 2 - 0 + (2n+1)\pi \cdot (-1) = 2 - (2n+1)\pi$$. For $$n \geq 1$$, this gives $$f'(x_n) = 2 - 3\pi < 0$$.

Similarly, at $$x_n = \frac{1}{2n\pi}$$, we have $$\cos\left(\frac{1}{x_n}\right) = 1$$ and $$\sin\left(\frac{1}{x_n}\right) = 0$$, so $$f'(x_n) = 2 + 2n\pi > 0$$.

Since $$f'(x)$$ takes both positive and negative values for $$x$$ near $$0^+$$, the function is not monotonic on $$(0, \infty)$$.

For $$x < 0$$, $$|x| = -x$$, so $$f(x) = -\left(2 - \sin\left(\frac{1}{x}\right)\right)x$$. Differentiating: $$f'(x) = -2 + \sin\left(\frac{1}{x}\right) - \frac{1}{x}\cos\left(\frac{1}{x}\right)$$.

By the same argument, at $$x_n = -\frac{1}{2n\pi}$$ (approaching $$0$$ from the left), $$\cos\left(\frac{1}{x_n}\right) = 1$$ and $$\frac{1}{x_n} = -2n\pi$$, so $$f'(x_n) = -2 + 0 - (-2n\pi)(1) = -2 + 2n\pi > 0$$ for $$n \geq 1$$. At $$x_n = -\frac{1}{(2n+1)\pi}$$, $$\cos\left(\frac{1}{x_n}\right) = -1$$ and $$\frac{1}{x_n} = -(2n+1)\pi$$, so $$f'(x_n) = -2 + 0 - (-(2n+1)\pi)(-1) = -2 - (2n+1)\pi < 0$$.

So $$f'(x)$$ also changes sign near $$0^-$$, meaning $$f$$ is not monotonic on $$(-\infty, 0)$$ either.

Therefore, $$f$$ is not monotonic on $$(-\infty, 0)$$ and $$(0, \infty)$$, which is Option B.

The curve is $$y = x^3$$. At the point $$P(t, t^3)$$, the slope of the tangent is $$\frac{dy}{dx} = 3t^2$$.

The equation of the tangent at $$P$$ is $$y - t^3 = 3t^2(x - t)$$, which simplifies to $$y = 3t^2 x - 2t^3$$.

To find the other point of intersection $$Q$$, we solve $$x^3 = 3t^2 x - 2t^3$$, giving $$x^3 - 3t^2 x + 2t^3 = 0$$.

Since $$x = t$$ is a repeated root (tangent touches at $$P$$), we factor: $$(x - t)^2(x + 2t) = 0$$.

So $$Q = (-2t, (-2t)^3) = (-2t, -8t^3)$$.

The point dividing $$PQ$$ internally in the ratio $$1 : 2$$ has ordinate $$\frac{1 \times (-8t^3) + 2 \times t^3}{1 + 2} = \frac{-8t^3 + 2t^3}{3} = \frac{-6t^3}{3} = -2t^3$$.

Hence, the correct answer is Option B.

A function is increasing wherever its derivative is positive. We compute $$f'(x)$$ on each piece separately.

For $$x > 0$$: $$f(x) = -\frac{4}{3}x^3 + 2x^2 + 3x$$, so $$f'(x) = -4x^2 + 4x + 3$$. Setting $$f'(x) > 0$$ gives $$4x^2 - 4x - 3 < 0$$, which factors as $$(2x - 3)(2x + 1) < 0$$. This holds for $$-\frac{1}{2} < x < \frac{3}{2}$$. Since we are on the region $$x > 0$$, the derivative is positive for $$0 < x < \frac{3}{2}$$.

For $$x \le 0$$: $$f(x) = 3x e^x$$, so $$f'(x) = 3e^x + 3x e^x = 3e^x(1 + x)$$. Since $$e^x > 0$$ always, $$f'(x) > 0$$ when $$1 + x > 0$$, i.e., $$x > -1$$. So the derivative is positive for $$-1 < x \le 0$$.

We must also verify continuity at $$x = 0$$: from the left, $$f(0) = 0$$; from the right, $$\lim_{x \to 0^+} f(x) = 0$$. So $$f$$ is continuous. Combining the two intervals, $$f$$ is increasing on $$(-1, 0] \cup (0, \frac{3}{2}) = \left(-1, \frac{3}{2}\right)$$.

The answer is $$\left(-1, \frac{3}{2}\right)$$.

We have the function

$$f(x)=3\sin^4x+10\sin^3x+6\sin^2x-3,$$

defined for $$x\in\left[-\dfrac{\pi}{6},\dfrac{\pi}{2}\right].$$ To study where the function is increasing or decreasing we need its first derivative.

First, recall the chain-rule differentiation formula:

$$\dfrac{d}{dx}\bigl[\sin^n x\bigr]=n\sin^{\,n-1}x\cdot\cos x.$$

Using this rule term-by-term we get

$$$\begin{aligned} \dfrac{d}{dx}\bigl[3\sin^4x\bigr] &=3\cdot4\sin^{3}x\cos x=12\sin^{3}x\cos x,\\[4pt] \dfrac{d}{dx}\bigl[10\sin^3x\bigr] &=10\cdot3\sin^{2}x\cos x=30\sin^{2}x\cos x,\\[4pt] \dfrac{d}{dx}\bigl[6\sin^2x\bigr] &=6\cdot2\sin x\cos x=12\sin x\cos x,\\[4pt] \dfrac{d}{dx}\bigl[-3\bigr] &=0. \end{aligned}$$$

Adding them, the derivative is

$$$\begin{aligned} f'(x)&=12\sin^{3}x\cos x+30\sin^{2}x\cos x+12\sin x\cos x\\[6pt] &=\cos x\bigl(12\sin^{3}x+30\sin^{2}x+12\sin x\bigr). \end{aligned}$$$

Now we factor the polynomial in $$\sin x$$ inside the parentheses. First take the common factor $$6\sin x$$:

$$$12\sin^{3}x+30\sin^{2}x+12\sin x =6\sin x\bigl(2\sin^{2}x+5\sin x+2\bigr).$$$

Hence

$$$f'(x)=6\sin x\;\cos x\;\bigl(2\sin^{2}x+5\sin x+2\bigr).$$$

We must study the sign of each factor on the interval $$\left(-\dfrac{\pi}{6},\dfrac{\pi}{2}\right).$$

Factor 1: $$\cos x.$$ For $$x\in\left(-\dfrac{\pi}{6},\dfrac{\pi}{2}\right)$$ we have $$\cos x>0$$ because the cosine is positive in the first and fourth quadrants up to $$\dfrac{\pi}{2}.$$

Factor 2: $$\sin x.$$ This changes sign at $$x=0:$$

• For $$x\in\left(-\dfrac{\pi}{6},0\right)\!,$$ we have $$\sin x<0.$$
• For $$x\in\left(0,\dfrac{\pi}{2}\right)\!,$$ we have $$\sin x>0.$$

Factor 3: $$2\sin^{2}x+5\sin x+2.$$ Put $$t=\sin x.$$ Then we analyse the quadratic

$$q(t)=2t^{2}+5t+2.$$ Its discriminant is $$\Delta=5^{2}-4\cdot2\cdot2=25-16=9,$$ so the roots are $$t=\dfrac{-5\pm\sqrt9}{2\cdot2}=\dfrac{-5\pm3}{4},$$ that is, $$t_{1}=-2,\qquad t_{2}=-\dfrac12.$$ Because the leading coefficient $$2>0,$$ the quadratic is positive outside the interval between its roots and negative inside it. Within the range $$-1\le t\le1$$ that $$\sin x$$ can attain, only the root $$t=-\dfrac12$$ is relevant. Consequently

• For $$\sin x\in\bigl(-1,-\dfrac12\bigr)$$ the quadratic is negative.
• For $$\sin x\in\bigl(-\dfrac12,1\bigr)$$ the quadratic is positive.

Note that $$\sin\bigl(-\tfrac{\pi}{6}\bigr)=-\dfrac12,$$ so on $$\left(-\dfrac{\pi}{6},0\right)$$ we actually have $$\sin x\in\bigl(-\dfrac12,0\bigr),$$ and therefore $$2\sin^{2}x+5\sin x+2>0$$ throughout the whole sub-interval.

Now we collect the signs interval-wise.

For $$x\in\left(-\dfrac{\pi}{6},0\right):$$

$$$\cos x>0,\qquad \sin x<0,\qquad 2\sin^{2}x+5\sin x+2>0.$$$

Multiplying, we get

$$f'(x)=\bigl(+\bigr)\bigl(-\bigr)\bigl(+\bigr)=\boxed{-}.$$

So $$f'(x)<0,$$ meaning $$f(x)$$ is strictly decreasing on $$\left(-\dfrac{\pi}{6},0\right).$$

For $$x\in\left(0,\dfrac{\pi}{2}\right):$$

$$$\cos x>0,\qquad \sin x>0,\qquad 2\sin^{2}x+5\sin x+2>0.$$$

Hence

$$f'(x)=\bigl(+\bigr)\bigl(+\bigr)\bigl(+\bigr)=\boxed{+},$$

so $$f(x)$$ is strictly increasing on $$\left(0,\dfrac{\pi}{2}\right).$$

Combining the two results:

• Decreasing on $$\left(-\dfrac{\pi}{6},0\right).$$
• Increasing on $$\left(0,\dfrac{\pi}{2}\right).$$

Among the given options, only Option D states that $$f$$ is decreasing on $$\left(-\dfrac{\pi}{6},0\right).$$

Hence, the correct answer is Option D.

Given,

$$f(x)=ax^2+6x-15$$

Then,

$$f'(x)=2ax+6$$

Since

$$f(x)$$ is increasing before

$$x=\frac34$$

and decreasing after it, the point $$x=\frac34$$ is a point of local maximum.

Hence,

$$f'\left(\frac34\right)=0$$

Therefore,

$$2a\left(\frac34\right)+6=0$$

$$\frac{3a}{2}=-6$$

$$a=-4$$

Now,

$$g(x)=ax^2-6x+15=-4x^2-6x+15$$

Then,

$$g'(x)=-8x-6$$

Setting $$g'(x)=0,$$

$$-8x-6=0$$

$$x=-\frac34$$

Also,

$$g''(x)=-8<0$$

Hence,

$$g(x)$$ has a local maximum at $$x=-\frac34$$

We have the function $$f(x)=\left(\dfrac{2}{x}\right)^{x^{2}},\;x\gt 0.$$

To locate a local maximum we first study its logarithm, because the natural logarithm is strictly increasing and therefore preserves the location of maxima.

Put $$y=\ln f(x).$$ Then

$$y=\ln\!\left(\left(\dfrac{2}{x}\right)^{x^{2}}\right)=x^{2}\,\ln\!\left(\dfrac{2}{x}\right).$$

Using the law $$\ln\!\left(\dfrac{a}{b}\right)=\ln a-\ln b,$$ we get

$$y=x^{2}\bigl(\ln 2-\ln x\bigr).$$

Now we differentiate $$y$$ with respect to $$x$$. Remember the product rule $$\dfrac{d}{dx}[u(x)\,v(x)]=u'(x)\,v(x)+u(x)\,v'(x).$$ Taking $$u=x^{2},\;v=\ln 2-\ln x,$$ we find

$$\dfrac{dy}{dx}=2x\bigl(\ln 2-\ln x\bigr)+x^{2}\Bigl(0-\dfrac{1}{x}\Bigr).$$

Simplifying step by step,

$$\dfrac{dy}{dx}=2x\ln 2-2x\ln x-x.$$

Factorising the common $$x$$ gives

$$\dfrac{dy}{dx}=x\bigl(2\ln 2-2\ln x-1\bigr).$$

For a critical point we set $$\dfrac{dy}{dx}=0.$$ Since $$x\gt 0$$, the factor $$x$$ cannot vanish, so we must have

$$2\ln 2-2\ln x-1=0.$$

Re-arranging,

$$2\bigl(\ln 2-\ln x\bigr)=1\;\;\Longrightarrow\;\;\ln\!\left(\dfrac{2}{x}\right)=\dfrac{1}{2}.$$

Exponentiating both sides with the base $$e$$ (using the rule $$\ln a=b\Longrightarrow a=e^{b}$$) we obtain

$$\dfrac{2}{x}=e^{1/2}=\sqrt{e},\quad\text{so}\quad x=\dfrac{2}{\sqrt{e}}.$$

To confirm that this critical point gives a maximum, we compute the second derivative. Starting from

$$\dfrac{dy}{dx}=x\bigl(2\ln 2-2\ln x-1\bigr),$$

differentiate again with the product rule:

$$\dfrac{d^{2}y}{dx^{2}}=1\cdot\bigl(2\ln 2-2\ln x-1\bigr)+x\Bigl(-\dfrac{2}{x}\Bigr).$$

Simplifying,

$$\dfrac{d^{2}y}{dx^{2}}=2\ln 2-2\ln x-1-2=2\ln 2-2\ln x-3.$$

At $$x=\dfrac{2}{\sqrt{e}}$$ we have $$\ln x=\ln 2-\dfrac{1}{2},$$ hence

$$\dfrac{d^{2}y}{dx^{2}}\Bigg|_{x=\frac{2}{\sqrt{e}}}=2\ln 2-2\Bigl(\ln 2-\dfrac{1}{2}\Bigr)-3 =2\ln 2-2\ln 2+1-3=-2\lt 0.$$

Because the second derivative is negative, the point is indeed a local maximum.

Now we compute the maximum value itself. Substitute $$x=\dfrac{2}{\sqrt{e}}$$ into the original function:

$$f_{\text{max}}=\left(\dfrac{2}{\dfrac{2}{\sqrt{e}}}\right)^{\left(\dfrac{2}{\sqrt{e}}\right)^{2}} =\bigl(\sqrt{e}\bigr)^{\frac{4}{e}}.$$

Recall that $$\sqrt{e}=e^{1/2}.$$ Therefore

$$f_{\text{max}}=\left(e^{1/2}\right)^{\frac{4}{e}}=e^{\frac{1}{2}\cdot\frac{4}{e}}=e^{\frac{2}{e}}.$$

This value matches Option C.

Hence, the correct answer is Option C.

We simplify the function first. Note that $$\cot\left(\frac{x}{2}\right)\sin^2\left(\frac{x}{2}\right) = \frac{\cos(x/2)}{\sin(x/2)} \cdot \sin^2\left(\frac{x}{2}\right) = \cos\left(\frac{x}{2}\right)\sin\left(\frac{x}{2}\right) = \frac{\sin x}{2}$$.

So the function becomes $$f(x) = (4a-3)(x + \log_e 5) + 2(a-7) \cdot \frac{\sin x}{2} = (4a-3)(x + \ln 5) + (a-7)\sin x$$.

Differentiating: $$f'(x) = (4a-3) + (a-7)\cos x$$. For critical points, we need $$f'(x) = 0$$, which gives $$\cos x = -\frac{4a-3}{a-7} = \frac{3-4a}{a-7}$$.

For this equation to have a solution, we need $$\left|\frac{3-4a}{a-7}\right| \leq 1$$, which means $$(3-4a)^2 \leq (a-7)^2$$. Expanding: $$9 - 24a + 16a^2 \leq a^2 - 14a + 49$$, which simplifies to $$15a^2 - 10a - 40 \leq 0$$, or $$3a^2 - 2a - 8 \leq 0$$.

Factoring: $$(3a + 4)(a - 2) \leq 0$$ (since roots are $$a = \frac{2 \pm \sqrt{4+96}}{6} = \frac{2 \pm 10}{6}$$, giving $$a = 2$$ or $$a = -\frac{4}{3}$$). This inequality holds for $$-\frac{4}{3} \leq a \leq 2$$.

Therefore, the range of $$a$$ is $$\left[-\frac{4}{3}, 2\right]$$.

The line is $$x - y = 1$$, or equivalently $$x - y - 1 = 0$$. The parabola is $$x^2 = 2y$$, so a general point on the parabola can be written as $$\left(t, \frac{t^2}{2}\right)$$.

The distance from this point to the line $$x - y - 1 = 0$$ is $$d(t) = \frac{\left|t - \frac{t^2}{2} - 1\right|}{\sqrt{1^2 + (-1)^2}} = \frac{\left|t - \frac{t^2}{2} - 1\right|}{\sqrt{2}}$$.

Let $$g(t) = t - \frac{t^2}{2} - 1$$. To find the extremum, we set $$g'(t) = 1 - t = 0$$, giving $$t = 1$$.

At $$t = 1$$: $$g(1) = 1 - \frac{1}{2} - 1 = -\frac{1}{2}$$. Since $$g''(t) = -1 < 0$$, this is the maximum of $$g(t)$$. Since $$g(t) \leq -\frac{1}{2} < 0$$ for all $$t$$, the minimum of $$|g(t)|$$ is $$\frac{1}{2}$$, occurring at $$t = 1$$.

The shortest distance is $$\frac{1/2}{\sqrt{2}} = \frac{1}{2\sqrt{2}}$$.

We are given a twice-differentiable function $$f : (a,\,b) \to \mathbb R$$ defined by the relation

$$f(x)=\int_a^{x} g(t)\,dt,$$

for some differentiable function $$g(x).$$

Because $$f(x)$$ is an integral of $$g(t)$$, we immediately obtain its first and second derivatives with the help of the Fundamental Theorem of Calculus:

First derivative formula: If $$F(x)=\displaystyle\int_a^{x} h(t)\,dt,$$ then $$F'(x)=h(x).$$ Hence,

$$f'(x)=g(x).$$

Second derivative formula: Differentiating once more, we have

$$f''(x)=g'(x).$$

So the chain of equalities we have is

$$f'(x)=g(x)\quad\text{and}\quad f''(x)=g'(x).$$

The equation $$f(x)=0$$ is said to possess exactly five distinct roots lying strictly inside the open interval $$(a,\,b).$$ Let us denote these five roots in increasing order by

$$a<\alpha_1<\alpha_2<\alpha_3<\alpha_4<\alpha_5<b.$$

Now we invoke Rolle’s Theorem. The theorem states:

Rolle’s Theorem (verbatim): If a function $$H(x)$$ is continuous on a closed interval $$[p,\,q]$$, differentiable on the open interval $$(p,\,q),$$ and satisfies $$H(p)=H(q),$$ then there exists at least one point $$c\in(p,\,q)$$ such that $$H'(c)=0.$$

We apply this theorem to $$f(x)$$ on every pair of consecutive zeros. Observe that

$$f(\alpha_k)=0=f(\alpha_{k+1})\quad\text{for }k=1,2,3,4.$$

Because $$f(x)$$ is continuous and differentiable on each closed interval $$[\alpha_k,\,\alpha_{k+1}],$$ Rolle’s theorem guarantees the existence of at least one number $$\beta_k\in(\alpha_k,\,\alpha_{k+1})$$ for which

$$f'(\beta_k)=0.$$

Recall that $$f'(x)=g(x).$$ Therefore, each $$\beta_k$$ is actually a zero of $$g(x).$$ We have located at least

$$\beta_1,\;\beta_2,\;\beta_3,\;\beta_4$$

with

$$\alpha_1<\beta_1<\alpha_2<\beta_2<\alpha_3<\beta_3<\alpha_4<\beta_4<\alpha_5.$$

Hence $$g(x)=0$$ possesses at least four distinct roots inside $$(a,\,b).$$ Summarising:

Number of distinct roots of $$g(x)=0$$ is at least $$4.$$

Next, we examine the derivative $$g'(x).$$ Because $$g(x)$$ is differentiable everywhere on $$(a,\,b),$$ we may again apply Rolle’s theorem, but now to $$g(x)$$ itself. Arrange its (at least four) distinct zeros in increasing order, say

$$\beta_1<\beta_2<\beta_3<\beta_4.$$

On each interval $$[\beta_j,\,\beta_{j+1}]$$ for $$j=1,2,3,$$ the conditions of Rolle’s theorem are satisfied for $$g(x)$$ (continuity, differentiability, and equal end-values $$g(\beta_j)=g(\beta_{j+1})=0$$). Consequently, for every such pair there is a point $$\gamma_j\in(\beta_j,\,\beta_{j+1})$$ where

$$g'(\gamma_j)=0.$$

Thus we have uncovered at least

$$\gamma_1,\;\gamma_2,\;\gamma_3$$

satisfying

$$\beta_1<\gamma_1<\beta_2<\gamma_2<\beta_3<\gamma_3<\beta_4.$$

Observe that each $$\gamma_j$$ lies strictly between neighbouring zeros $$\beta_j$$ and $$\beta_{j+1}$$ of $$g(x).$$ Hence no $$\gamma_j$$ coincides with any $$\beta_k.$$

We now collect all the roots relevant to the equation $$g(x)g'(x)=0.$$ This equation factors as

$$g(x)g'(x)=0\quad\iff\quad g(x)=0\quad\text{or}\quad g'(x)=0.$$

The roots we have already located are

• $$\beta_1,\;\beta_2,\;\beta_3,\;\beta_4$$ where $$g(x)=0,$$ and

• $$\gamma_1,\;\gamma_2,\;\gamma_3$$ where $$g'(x)=0.$$

Because each $$\gamma_j$$ sits strictly between two distinct $$\beta_k$$’s, none of the seven points coincide. Therefore, the set of distinct solutions of $$g(x)g'(x)=0$$ inside $$(a,\,b)$$ contains at least

$$4+3=7$$

elements.

Consequently, the equation $$g(x)g'(x)=0$$ has at least seven distinct roots in the open interval $$(a,\,b).$$

Hence, the correct answer is Option C.

Given:

Side of equilateral triangle,

$$s=2\sqrt2$$

Height of the triangle:

$$H=\frac{\sqrt3}{2}s$$

$$=\frac{\sqrt3}{2}(2\sqrt2)$$

$$=\sqrt6$$

Base of triangle:

$$B=2\sqrt2$$

Let the inscribed rectangle have:

- width $$w$$

- height $$h$$

Using similar triangles,

$$\frac{w}{B}=\frac{H-h}{H}$$

Hence,

$$w=B\left(1-\frac{h}{H}\right)$$

Area of rectangle:

$$A=wh$$

$$=Bh\left(1-\frac{h}{H}\right)$$

$$=B\left(h-\frac{h^2}{H}\right)$$

Differentiate with respect to $$h$$ :

$$\frac{dA}{dh}=B\left(1-\frac{2h}{H}\right)$$

For maximum area,

$$\frac{dA}{dh}=0$$

$$1-\frac{2h}{H}=0$$

$$h=\frac{H}{2}$$

Now,

$$w=B\left(1-\frac12\right)=\frac{B}{2}$$

Therefore maximum area is

$$A_{\max}=\frac{B}{2}\cdot\frac{H}{2}$$

$$=\frac{BH}{4}$$

But,

$$\frac12 BH$$

is the area of the triangle.

Hence,

$$A_{\max}=\frac12(\text{Area of triangle})$$

Area of equilateral triangle:

$$=\frac{\sqrt3}{4}s^2$$

$$=\frac{\sqrt3}{4}(2\sqrt2)^2$$

$$=\frac{\sqrt3}{4}\cdot8$$

$$=2\sqrt3$$

Thus,

$$A_{\max}=\frac{2\sqrt3}{2}$$

$$=\sqrt3$$

Therefore,

$$(A_{\max})^2=(\sqrt3)^2$$

$$=3$$

Hence, the required answer is

$$\boxed{3}$$

We are given the quadratic function $$f(x)=x^{2}+ax+1$$ and we have to study its monotonicity on the closed interval $$[1,\,2]$$.

First, recall the basic calculus fact: a differentiable function is increasing on an interval if and only if its derivative is non-negative throughout that interval, and it is decreasing if and only if its derivative is non-positive throughout that interval.

So we begin by finding the derivative. Using the standard power rule $$\frac{d}{dx}(x^{n})=nx^{n-1}$$ and the rule $$\frac{d}{dx}(kx)=k$$ for a constant $$k$$, we differentiate term by term:

$$f'(x)=\frac{d}{dx}(x^{2})+\frac{d}{dx}(ax)+\frac{d}{dx}(1)=2x+a+0=2x+a.$$

Observe that $$f'(x)=2x+a$$ is a linear function in $$x$$ with positive coefficient $$2$$ in front of $$x$$, so $$f'(x)$$ itself is strictly increasing with respect to $$x$$.

Now we treat the two required situations separately.

First, we want $$f(x)$$ to be increasing on $$[1,2]$$. Because $$f'(x)$$ is increasing, its minimum on the interval $$[1,2]$$ is attained at the left end point $$x=1$$. Therefore, the condition for non-negativity of the derivative on the whole interval is

$$f'(1)\ge 0\quad\Longrightarrow\quad 2\cdot1+a\ge 0\quad\Longrightarrow\quad a\ge -2.$$

We are asked for the least value of $$a$$ that makes the function increasing, so we take the boundary value

$$R=-2.$$

Next, we want $$f(x)$$ to be decreasing on $$[1,2]$$. Again, since $$f'(x)=2x+a$$ is increasing in $$x$$, its maximum on $$[1,2]$$ is attained at the right end point $$x=2$$. For the derivative to be non-positive everywhere in the interval we impose

$$f'(2)\le 0\quad\Longrightarrow\quad 2\cdot2+a\le 0\quad\Longrightarrow\quad a\le -4.$$

We need the greatest value of $$a$$ that fulfils this inequality, hence we choose the boundary value

$$S=-4.$$

Finally, we compute the required absolute difference:

$$|R-S|=\left|-2-(-4)\right|=\left|\,(-2)+4\,\right|=\left|\,2\,\right|=2.$$

Hence, the correct answer is Option 2.

We need to find where $$x = y^4$$ and $$xy = k$$ intersect at right angles. Differentiating $$x = y^4$$ with respect to $$x$$: $$1 = 4y^3 \frac{dy}{dx}$$, so $$\frac{dy}{dx} = \frac{1}{4y^3}$$.

Differentiating $$xy = k$$: $$y + x\frac{dy}{dx} = 0$$, so $$\frac{dy}{dx} = -\frac{y}{x}$$.

For the curves to cut at right angles, the product of their slopes must be $$-1$$: $$\frac{1}{4y^3} \times \left(-\frac{y}{x}\right) = -1$$. This simplifies to $$\frac{1}{4y^2 x} = 1$$, giving $$4y^2 x = 1$$.

At the point of intersection, $$x = y^4$$. Substituting: $$4y^2 \cdot y^4 = 1$$, so $$4y^6 = 1$$, which gives $$y^6 = \frac{1}{4}$$.

Also, $$k = xy = y^4 \cdot y = y^5$$, so $$k^6 = y^{30} = (y^6)^5 = \left(\frac{1}{4}\right)^5 = \frac{1}{1024}$$.

Therefore, $$(4k)^6 = 4^6 \cdot k^6 = 4096 \times \frac{1}{1024} = 4$$.

We have $$f(x) = ax^2 + bx + c$$ with $$f(-1) = a - b + c = 2$$, $$f'(x) = 2ax + b$$ so $$f'(-1) = -2a + b = 1$$, and $$f''(x) = 2a$$ with maximum value $$\frac{1}{2}$$ on $$(-1,1)$$, meaning $$2a \leq \frac{1}{2}$$, so $$a \leq \frac{1}{4}$$.

From $$f'(-1) = 1$$: $$b = 1 + 2a$$. From $$f(-1) = 2$$: $$a - (1+2a) + c = 2$$, giving $$c = 3 + a$$.

So $$f(x) = ax^2 + (1+2a)x + (3+a)$$. To find the maximum of $$f(x)$$ on $$[-1, 1]$$, we evaluate $$f(1) = a + 1 + 2a + 3 + a = 4a + 4$$.

Since $$f$$ is a quadratic with leading coefficient $$a \leq \frac{1}{4}$$, and the vertex is at $$x = -\frac{b}{2a} = -\frac{1+2a}{2a}$$, we need to check whether the maximum on $$[-1,1]$$ occurs at $$x = 1$$ or at the vertex.

For small positive $$a$$, the vertex $$x = -\frac{1+2a}{2a}$$ is very negative (far left of $$[-1,1]$$), so $$f$$ is increasing on $$[-1,1]$$ and the maximum is at $$x = 1$$. For $$a \leq 0$$, the parabola opens downward, and the vertex could be inside $$[-1,1]$$.

To maximize $$\alpha$$ (the maximum of $$f$$ on $$[-1,1]$$), we want to choose $$a$$ as large as possible since $$f(1) = 4a + 4$$ increases with $$a$$. The constraint gives $$a \leq \frac{1}{4}$$, and equality is achieved when $$f''(x) = \frac{1}{2}$$.

With $$a = \frac{1}{4}$$: $$b = 1 + \frac{1}{2} = \frac{3}{2}$$, $$c = 3 + \frac{1}{4} = \frac{13}{4}$$. The vertex is at $$x = -\frac{3/2}{1/2} = -3$$, which is outside $$[-1,1]$$, so $$f$$ is increasing on $$[-1,1]$$.

The maximum value is $$f(1) = 4 \cdot \frac{1}{4} + 4 = 5$$. Therefore the least value of $$\alpha$$ such that $$f(x) \leq \alpha$$ for all $$x \in [-1,1]$$ is $$\alpha = 5$$.

We maximize $$z = 6xy + y^2$$ subject to $$3x + 4y \leq 100$$, $$4x + 3y \leq 75$$, $$x \geq 0$$, $$y \geq 0$$.

First we find the corner points of the feasible region. The constraints $$3x + 4y = 100$$ and $$4x + 3y = 75$$ intersect where: multiplying the first by 4 and the second by 3 gives $$12x + 16y = 400$$ and $$12x + 9y = 225$$. Subtracting: $$7y = 175$$, so $$y = 25$$ and $$x = 0$$. The feasible region has corners at $$(0,0)$$, $$(75/4, 0) = (18.75, 0)$$, and $$(0, 25)$$.

Evaluating $$z$$ at the corners: at $$(0,0)$$: $$z = 0$$. At $$(18.75, 0)$$: $$z = 0$$. At $$(0, 25)$$: $$z = 625$$.

Since $$z = 6xy + y^2$$ is not linear, we must also check the boundary edges. On the axes ($$x = 0$$ or $$y = 0$$), $$z$$ simplifies to $$y^2$$ or 0, so the maximum on these boundaries is at $$(0, 25)$$ with $$z = 625$$.

Now check the edge $$4x + 3y = 75$$ with $$0 \leq y \leq 25$$. Here $$x = \frac{75 - 3y}{4}$$. Substituting into $$z$$:

$$z = 6 \cdot \frac{75-3y}{4} \cdot y + y^2 = \frac{6y(75-3y)}{4} + y^2 = \frac{450y - 18y^2}{4} + y^2 = \frac{450y - 18y^2 + 4y^2}{4} = \frac{450y - 14y^2}{4}$$

Taking the derivative: $$\frac{dz}{dy} = \frac{450 - 28y}{4}$$. Setting this to zero: $$y = \frac{450}{28} = \frac{225}{14} \approx 16.07$$. The second derivative is $$\frac{-28}{4} = -7 < 0$$, confirming this is a maximum.

At $$y = \frac{225}{14}$$: $$x = \frac{75 - 675/14}{4} = \frac{375/14}{4} = \frac{375}{56} \approx 6.70$$. We verify $$3x + 4y = \frac{1125}{56} + \frac{900}{14} = \frac{1125 + 3600}{56} = \frac{4725}{56} \approx 84.4 \leq 100$$, so the point is feasible.

$$z = \frac{450 \cdot 225/14 - 14 \cdot (225/14)^2}{4} = \frac{101250/14 - 50625/14}{4} = \frac{50625}{56} \approx 904$$.

Comparing all candidates: corners give at most 625, while the boundary maximum gives $$\frac{50625}{56} \approx 904$$.

The maximum value of $$z$$ is 904.

Let us assume that the required cubic polynomial is of the general form

$$f(x)=ax^{3}+bx^{2}+cx+d,$$

where $$a,\;b,\;c,\;d$$ are real constants that we have to determine.

We are given the two functional values

$$f(1)=-10 \quad\text{and}\quad f(-1)=6.$$

We are also told that the function has a local minimum at $$x=1$$. For a local extremum we know the basic calculus fact that

$$f'(x)=0 \quad\text{at that point.}$$

So we first compute the derivative:

$$f'(x)=\frac{d}{dx}\bigl(ax^{3}+bx^{2}+cx+d\bigr)=3ax^{2}+2bx+c.$$

Using the extremum condition at $$x=1$$, we write

$$f'(1)=3a(1)^{2}+2b(1)+c=3a+2b+c=0.$$ So we obtain our first equation

$$3a+2b+c=0.\qquad(1)$$

Next, we are told that $$f'(x)$$ itself (which is a quadratic) has a local minimum at $$x=-1$$. Again applying the same extremum rule to $$f'(x)$$, we differentiate once more:

$$f''(x)=\frac{d}{dx}\bigl(3ax^{2}+2bx+c\bigr)=6ax+2b.$$

A local extremum of $$f'(x)$$ at $$x=-1$$ means

$$f''(-1)=0.$$

Substituting $$x=-1$$ in the expression of $$f''(x)$$ we get

$$6a(-1)+2b=0\;\Longrightarrow\;-6a+2b=0.$$

Simplifying, we arrive at

$$2b=6a \;\Longrightarrow\; b=3a.\qquad(2)$$

Now we return to equation (1) and replace $$b$$ with $$3a$$ from equation (2):

$$3a+2(3a)+c=0 \;\Longrightarrow\; 3a+6a+c=0 \;\Longrightarrow\; 9a+c=0.$$

Thus we get

$$c=-9a.\qquad(3)$$

We have expressed $$b$$ and $$c$$ in terms of $$a$$. The constant $$d$$ can be found from the given value $$f(1)=-10$$. Substituting $$x=1$$ in the general expression of $$f(x)$$, we have

$$f(1)=a(1)^{3}+b(1)^{2}+c(1)+d=a+b+c+d=-10.$$

Replacing $$b$$ and $$c$$ by their expressions from (2) and (3):

$$a+3a-9a+d=-10.$$

Combining like terms,

$$(-5a)+d=-10 \;\Longrightarrow\; d=-10+5a.\qquad(4)$$

The last given condition is $$f(-1)=6.$$ Putting $$x=-1$$ into $$f(x)$$ gives

$$f(-1)=a(-1)^{3}+b(-1)^{2}+c(-1)+d=-a+b-c+d=6.$$

Again substitute $$b=3a,\;c=-9a,\;d=-10+5a$$:

$$-a+3a-(-9a)+(-10+5a)=6.$$

Observe that $$-(-9a)=+9a$$, so the left-hand side simplifies to

$$(-a+3a+9a+5a)-10 = (16a)-10.$$

Hence we get

$$16a-10=6 \;\Longrightarrow\; 16a=16 \;\Longrightarrow\; a=1.$$

With $$a=1$$ known, the remaining coefficients follow:

$$b=3a=3,\quad c=-9a=-9,\quad d=-10+5a=-10+5=-5.$$

Therefore the fully determined cubic polynomial is

$$f(x)=x^{3}+3x^{2}-9x-5.$$

Finally, we are asked to compute $$f(3)$$. Substituting $$x=3$$ gives

$$f(3)=3^{3}+3\cdot3^{2}-9\cdot3-5=27+27-27-5.$$

Combining term by term,

$$27+27=54,\qquad 54-27=27,\qquad 27-5=22.$$

So, the answer is $$22$$.

Let $$P(x) = ax^3 + bx^2 + cx + d$$ be a cubic polynomial with $$P(-3) = 0$$. Since $$P$$ has a local minimum at $$x = 1$$ and a local maximum at $$x = -1$$, we need $$P'(1) = 0$$ and $$P'(-1) = 0$$, where $$P'(x) = 3ax^2 + 2bx + c$$.

From $$P'(1) = 3a + 2b + c = 0$$ and $$P'(-1) = 3a - 2b + c = 0$$, subtracting gives $$4b = 0$$, so $$b = 0$$. Then $$c = -3a$$. With $$b = 0$$, the polynomial is $$P(x) = ax^3 - 3ax + d$$.

Applying $$P(-3) = 0$$: $$a(-27) - 3a(-3) + d = -27a + 9a + d = -18a + d = 0$$, so $$d = 18a$$. Thus $$P(x) = a(x^3 - 3x + 18)$$.

Now we use $$\int_{-1}^{1}P(x)\,dx = 18$$. Since $$x^3$$ and $$x$$ are odd functions, their integrals over $$[-1,1]$$ vanish, leaving $$\int_{-1}^{1}18a\,dx = 18a \cdot 2 = 36a = 18$$, giving $$a = \dfrac{1}{2}$$.

The sum of all coefficients of $$P(x)$$ is $$P(1) = \dfrac{1}{2}(1 - 3 + 18) = \dfrac{16}{2} = 8$$.

Let the total length of the wire be 36 m. Suppose $$k$$ metres of the wire are used to form the circle, so the remaining $$36-k$$ metres form the square.

If the side of the square is $$x$$, we have the perimeter-side relation for a square: $$4x = 36-k$$, therefore

$$x = \dfrac{36-k}{4}.$$

The circle whose circumference is $$k$$ has radius $$r$$ given by the circumference formula $$2\pi r = k$$, so

$$r = \dfrac{k}{2\pi}.$$

Now, write the total area $$A$$ of the two figures. The area of the square is $$x^{2}$$ and the area of the circle is $$\pi r^{2}$$. Thus

$$A = x^{2} + \pi r^{2}.$$

Substituting the expressions for $$x$$ and $$r$$, we get

$$A = \left(\dfrac{36-k}{4}\right)^{2} + \pi\left(\dfrac{k}{2\pi}\right)^{2}.$$

Simplifying the second term first:

$$\pi\left(\dfrac{k}{2\pi}\right)^{2} = \pi \cdot \dfrac{k^{2}}{4\pi^{2}} = \dfrac{k^{2}}{4\pi}.$$

Therefore

$$A(k) = \left(\dfrac{36-k}{4}\right)^{2} + \dfrac{k^{2}}{4\pi}.$$

To minimise the area, differentiate $$A$$ with respect to $$k$$ and set the derivative equal to zero.

First expand the square term and differentiate directly:

$$A(k) = \dfrac{(36-k)^{2}}{16} + \dfrac{k^{2}}{4\pi}.$$

Differentiate term-by-term:

$$\dfrac{dA}{dk} = \dfrac{1}{16}\cdot 2(36-k)(-1) + \dfrac{1}{4\pi}\cdot 2k.$$

Simplify each part:

$$\dfrac{dA}{dk} = -\dfrac{36-k}{8} + \dfrac{k}{2\pi}.$$

Set the derivative to zero for a minimum:

$$-\dfrac{36-k}{8} + \dfrac{k}{2\pi} = 0.$$

Multiply through by the common denominator $$8\pi$$ to clear fractions:

$$-\pi(36-k) + 4k = 0.$$

Expand the first product:

$$-36\pi + \pi k + 4k = 0.$$

Group the $$k$$ terms:

$$(\pi + 4)k = 36\pi.$$

Solve for $$k$$:

$$k = \dfrac{36\pi}{\pi + 4}.$$

The expression asked for in the problem is $$\left(\dfrac{4}{\pi} + 1\right)k$$. Compute it by direct substitution:

$$\left(\dfrac{4}{\pi} + 1\right)k = \left(\dfrac{4 + \pi}{\pi}\right)\left(\dfrac{36\pi}{\pi + 4}\right).$$

The numerator and denominator each contain the common factor $$\pi + 4$$, which cancels neatly with $$4 + \pi$$, leaving

$$\left(\dfrac{4 + \pi}{\pi}\right)\left(\dfrac{36\pi}{\pi + 4}\right) = 36.$$

So, the answer is $$36$$.

We have $$y(x) = \int_0^x (2t^2 - 15t + 10)\,dt$$. By the Fundamental Theorem of Calculus, $$y'(x) = 2x^2 - 15x + 10$$.

The normal to the curve at $$(a, b)$$ is parallel to $$x + 3y = -5$$, whose slope is $$-\frac{1}{3}$$. The slope of the normal is $$-\frac{1}{y'(a)}$$, so $$-\frac{1}{y'(a)} = -\frac{1}{3}$$, giving $$y'(a) = 3$$.

Setting $$2a^2 - 15a + 10 = 3$$: $$2a^2 - 15a + 7 = 0$$. Using the quadratic formula: $$a = \frac{15 \pm \sqrt{225 - 56}}{4} = \frac{15 \pm \sqrt{169}}{4} = \frac{15 \pm 13}{4}$$.

So $$a = 7$$ or $$a = \frac{1}{2}$$. Since $$a > 1$$, we have $$a = 7$$.

Now $$b = y(7) = \int_0^7 (2t^2 - 15t + 10)\,dt = \left[\frac{2t^3}{3} - \frac{15t^2}{2} + 10t\right]_0^7$$.

Evaluating: $$\frac{2(343)}{3} - \frac{15(49)}{2} + 70 = \frac{686}{3} - \frac{735}{2} + 70$$.

Finding a common denominator of 6: $$\frac{1372}{6} - \frac{2205}{6} + \frac{420}{6} = \frac{1372 - 2205 + 420}{6} = \frac{-413}{6}$$.

Therefore $$|a + 6b| = \left|7 + 6 \cdot \frac{-413}{6}\right| = |7 - 413| = |-406| = 406$$.

We have the cubic function $$f(x)=2x^3-3x^2-12x.$$

To locate its turning points we first differentiate. The rule is: for any power term $$kx^n,$$ the derivative is $$knx^{\,n-1}.$$ So,

$$f'(x)=\dfrac{d}{dx}(2x^3)-\dfrac{d}{dx}(3x^2)-\dfrac{d}{dx}(12x)$$ $$=2\cdot3x^{\,2}-3\cdot2x^{\,1}-12$$ $$=6x^2-6x-12.$$

We set the derivative equal to zero to find critical points:

$$6x^2-6x-12=0 \Longrightarrow 6\bigl(x^2-x-2\bigr)=0$$ $$x^2-x-2=0.$$

Factoring, $$x^2-x-2=(x-2)(x+1)=0,$$ hence

$$x=-1 \quad\text{or}\quad x=2.$$

Next we apply the second-derivative test. Differentiate once more: $$f''(x)=\dfrac{d}{dx}(6x^2-6x-12)=12x-6.$$

At $$x=-1$$ we get $$f''(-1)=12(-1)-6=-18<0,$$ indicating a local maximum. Thus $$a=-1.$$

At $$x=2$$ we get $$f''(2)=12(2)-6=18>0,$$ indicating a local minimum. Thus $$b=2.$$

The required region is bounded by the curve $$y=f(x),$$ the $$x$$-axis, and the vertical lines $$x=a=-1$$ and $$x=b=2.$$ Because the curve crosses the axis inside this interval, we must separate the area into parts that lie above and below the axis and use absolute values.

First we identify where $$f(x)=0.$$ Factorising, $$f(x)=x\bigl(2x^2-3x-12\bigr).$$ The obvious root is $$x=0.$$ (The other two roots are outside the present calculation, but they confirm that the graph crosses the axis only at $$x=0$$ between $$-1$$ and $$2.$$)

Hence we split the integral at $$x=0$$:

$$A=\int_{-1}^{0}f(x)\,dx-\int_{0}^{2}f(x)\,dx,$$ the minus sign in the second term ensuring positivity of area below the axis.

We now find an antiderivative. Using $$\int x^n\,dx=\dfrac{x^{n+1}}{n+1},$$

$$\int f(x)\,dx=\int(2x^3-3x^2-12x)\,dx$$ $$=2\cdot\dfrac{x^{4}}{4}-3\cdot\dfrac{x^{3}}{3}-12\cdot\dfrac{x^{2}}{2}+C$$ $$=\dfrac{x^{4}}{2}-x^{3}-6x^{2}+C.$$

Denote $$F(x)=\dfrac{x^{4}}{2}-x^{3}-6x^{2}.$$

Area above the axis (from $$-1$$ to $$0$$):

$$\int_{-1}^{0}f(x)\,dx=F(0)-F(-1).$$ Compute $$F(0)=0,$$ and $$F(-1)=\dfrac{(-1)^{4}}{2}-(-1)^{3}-6(-1)^{2}=\dfrac{1}{2}+1-6=-\dfrac{9}{2}.$$ So

$$\int_{-1}^{0}f(x)\,dx=0-\left(-\dfrac{9}{2}\right)=\dfrac{9}{2}=4.5.$$

Area below the axis (from $$0$$ to $$2$$):

$$\int_{0}^{2}f(x)\,dx=F(2)-F(0).$$ Compute $$F(2)=\dfrac{2^{4}}{2}-2^{3}-6\cdot2^{2}=8-8-24=-24.$$ Therefore

$$\int_{0}^{2}f(x)\,dx=-24.$$ Taking its absolute value gives an area of $$24.$$

Adding the two parts,

$$A=4.5+24=28.5=\dfrac{57}{2}.$$

Finally, the question asks for $$4A:$$

$$4A=4\times28.5=114.$$

Hence, the correct answer is Option 114.

Let $$f(x)$$ be a polynomial of degree 6 with leading coefficient 1. Since $$\lim_{x \to 0}\frac{f(x)}{x^3} = 1$$, the polynomial $$f(x)$$ must have no constant term, no $$x$$ term, and no $$x^2$$ term (otherwise the limit would be undefined or infinite), and the coefficient of $$x^3$$ must be 1.

So $$f(x) = x^6 + ax^5 + bx^4 + x^3$$ for some constants $$a$$ and $$b$$.

Since $$f(x)$$ has extrema at $$x = -1$$ and $$x = 1$$, we need $$f'(-1) = 0$$ and $$f'(1) = 0$$.

Computing the derivative: $$f'(x) = 6x^5 + 5ax^4 + 4bx^3 + 3x^2$$.

Setting $$f'(1) = 0$$: $$6 + 5a + 4b + 3 = 0$$, so $$5a + 4b = -9$$.

Setting $$f'(-1) = 0$$: $$-6 + 5a - 4b + 3 = 0$$, so $$5a - 4b = 3$$.

Adding these two equations: $$10a = -6$$, giving $$a = -\frac{3}{5}$$. Subtracting: $$8b = -12$$, giving $$b = -\frac{3}{2}$$.

Therefore, $$f(x) = x^6 - \frac{3}{5}x^5 - \frac{3}{2}x^4 + x^3$$.

Computing $$f(2) = 64 - \frac{3}{5}(32) - \frac{3}{2}(16) + 8 = 64 - \frac{96}{5} - 24 + 8 = 48 - \frac{96}{5} = \frac{240 - 96}{5} = \frac{144}{5}$$.

Therefore, $$5 \cdot f(2) = 5 \times \frac{144}{5} = 144$$.

We begin with the quartic equation

$$3x^{4}+4x^{3}-12x^{2}+4=0.$$

To determine how many real roots it possesses, we try to factor it into a product of two quadratic polynomials having real coefficients. Let us assume

$$3x^{4}+4x^{3}-12x^{2}+4=(x^{2}+ax+b)\,(3x^{2}+cx+d).$$

We now expand the right-hand side and equate coefficients term by term. Multiplying gives

$$\begin{aligned} (x^{2}+ax+b)\,(3x^{2}+cx+d) &=x^{2}(3x^{2}+cx+d)+ax(3x^{2}+cx+d)+b(3x^{2}+cx+d)\\ &=3x^{4}+cx^{3}+dx^{2}+3ax^{3}+acx^{2}+adx+3bx^{2}+bcx+bd\\ &=3x^{4}+(c+3a)x^{3}+(d+ac+3b)x^{2}+(ad+bc)x+bd. \end{aligned}$$

Comparing this with the original polynomial, we obtain the system

$$\begin{cases} c+3a=4,\\[4pt] d+ac+3b=-12,\\[4pt] ad+bc=0,\\[4pt] bd=4. \end{cases}$$

The last equation $$bd=4$$ suggests the integer possibilities $$(b,d)=(\pm1,\pm4)$$ or $$(\pm2,\pm2)$$. We test $$(b,d)=(-2,-2)$$ because it satisfies $$bd=4$$ and keeps the numbers small.

With $$b=-2$$ and $$d=-2$$, the third equation $$ad+bc=0$$ becomes

$$(-2)a+(-2)c=0 \;\Longrightarrow\; -2(a+c)=0 \;\Longrightarrow\; a+c=0 \;\Longrightarrow\; c=-a.$$

Substituting $$c=-a$$ into $$c+3a=4$$ gives

$$-a+3a=4 \;\Longrightarrow\; 2a=4 \;\Longrightarrow\; a=2,$$

and hence $$c=-2$$.

Finally, we check the second equation $$d+ac+3b=-12$$:

$$-2+(2)(-2)+3(-2)=-2-4-6=-12,$$

which is satisfied exactly. Thus our choice is consistent, and the factorisation is confirmed:

$$3x^{4}+4x^{3}-12x^{2}+4=(x^{2}+2x-2)\,(3x^{2}-2x-2).$$

Setting each quadratic factor equal to zero gives the roots.

For the first quadratic $$x^{2}+2x-2=0,$$ we apply the quadratic formula $$x=\dfrac{-2\pm\sqrt{(2)^{2}-4(1)(-2)}}{2(1)}=\dfrac{-2\pm\sqrt{4+8}}{2}=\dfrac{-2\pm\sqrt{12}}{2}=\dfrac{-2\pm2\sqrt{3}}{2}=-1\pm\sqrt{3}.$$

So we obtain two real roots $$x=-1+\sqrt{3}\quad\text{and}\quad x=-1-\sqrt{3}.$$

For the second quadratic $$3x^{2}-2x-2=0,$$ the quadratic formula gives

$$x=\dfrac{2\pm\sqrt{(-2)^{2}-4(3)(-2)}}{2\cdot3}=\dfrac{2\pm\sqrt{4+24}}{6}=\dfrac{2\pm\sqrt{28}}{6}=\dfrac{2\pm2\sqrt{7}}{6}=\dfrac{1\pm\sqrt{7}}{3}.$$

Thus we obtain two more real roots $$x=\dfrac{1+\sqrt{7}}{3}\quad\text{and}\quad x=\dfrac{1-\sqrt{7}}{3}.$$

All four solutions are real, and it is easy to see that they are distinct because the numerical values $$-1\pm\sqrt{3}$$ and $$(1\pm\sqrt{7})/3$$ are pairwise different.

Therefore the equation possesses four distinct real roots.

So, the answer is $$4$$.

Given :

• f is continuous on [0,1]
• f differentiable on (0,1)

So, we can apply Lagrange’s Mean Value Theorem (LMVT).

As we can say that f is also continuous on [c,1] and also differentiable on (c,1).

So we can apply LMVT on [c,1].

Apply LMVT on interval [c,1]:
since f satisfies conditions,

there exists some points let it be b such that :

   $$f'\left(c\right)\ =\ \frac{f\left(1\right)-f\left(c\right)}{1-c}$$

Hence the correct answer is

$$f'\left(c\right)\ =\ \frac{f\left(1\right)-f\left(c\right)}{1-c}$$

We have a function $$f$$ which is continuous on the closed interval $$[a,b]$$ and twice differentiable on the open interval $$(a,b)$$. For every point $$x$$ in $$(a,b)$$ it satisfies $$f'(x)>0$$ and $$f''(x)<0$$. The inequality $$f'(x)>0$$ tells us that the graph of $$f$$ rises as we move from left to right, while $$f''(x)<0$$ tells us that this rise is at a decreasing rate; that is, the first derivative $$f'$$ itself is strictly decreasing on $$(a,b)$$.

Fix any point $$c$$ with $$a<c<b$$ and consider the ratio

$$R=\frac{f(c)-f(a)}{f(b)-f(c)}.$$

Because $$f$$ is continuous on each closed sub-interval and differentiable on each open sub-interval, we can apply the Mean Value Theorem separately to the intervals $$[a,c]$$ and $$[c,b]$$.

Mean Value Theorem on the interval $$[a,c]$$. There exists some point $$x_{1}\in(a,c)$$ such that

$$f(c)-f(a)=f'(x_{1})(c-a).$$

Mean Value Theorem on the interval $$[c,b]$$. There exists some point $$x_{2}\in(c,b)$$ such that

$$f(b)-f(c)=f'(x_{2})(b-c).$$

Substituting these two Mean Value Theorem identities into the ratio $$R$$, we obtain

$$ R=\frac{f(c)-f(a)}{f(b)-f(c)} =\frac{f'(x_{1})(c-a)}{f'(x_{2})(b-c)}. $$

Now, observe the relative positions of the points: $$a<x_{1}<c<x_{2}<b.$$ Because $$f''(x)<0$$ on $$(a,b)$$, the first derivative $$f'(x)$$ is strictly decreasing. Hence, moving to the right decreases its value, so

$$f'(x_{1}) > f'(x_{2}).$$

Using this inequality we can compare $$R$$ with the ratio of the lengths of the sub-intervals:

$$ R =\frac{f'(x_{1})(c-a)}{f'(x_{2})(b-c)} > \frac{f'(x_{2})(c-a)}{f'(x_{2})(b-c)} =\frac{c-a}{\,b-c\,}. $$

Thus we have established the strict inequality

$$ \boxed{\displaystyle\frac{f(c)-f(a)}{f(b)-f(c)} \; >\; \frac{c-a}{\,b-c\,}}. $$

This inequality holds for every $$c\in(a,b)$$ whenever $$f'(x)>0$$ and $$f''(x)<0$$ on $$(a,b)$$. Looking at the options supplied, the right-hand side of this inequality is exactly the expression in Option D.

Hence, the correct answer is Option D.

We begin with the information that the polynomial $$f(x)$$ is of degree four and possesses the three critical points $$x=-1,\;0,\;1$$.

By definition, a critical point is a root of the first derivative. Hence we must have

$$f'(x)=0 \quad\text{when}\quad x=-1,\,0,\,1.$$

A fourth-degree polynomial has a third-degree derivative. Therefore the derivative can have at most three roots, and here all three are already specified. Thus those three roots account for the entire derivative, so the derivative must be proportional to the cubic factor that vanishes at these points:

$$f'(x)=k\,(x+1)\,x\,(x-1),$$

where $$k\neq0$$ is a constant of proportionality.

We next expand the cubic for convenience:

$$ (x+1)\,x\,(x-1)=x\,(x^2-1)=x^3-x. $$

So the derivative is explicitly

$$f'(x)=k\,(x^3-x).$$

To recover $$f(x)$$ itself we integrate. We recall the elementary antiderivatives:

$$\int x^3\,dx=\frac{x^4}{4},\qquad\int x\,dx=\frac{x^2}{2}.$$

Applying term-by-term integration and keeping the constant of integration $$C$$, we obtain

$$\begin{aligned} f(x) &= k\int (x^3-x)\,dx + C \\ &= k\left(\frac{x^4}{4}-\frac{x^2}{2}\right)+C. \end{aligned}$$

Thus

$$f(x)=\frac{k}{4}\,x^4-\frac{k}{2}\,x^2+C.$$

Now we form the set

$$T=\{x\in\mathbb R\mid f(x)=f(0)\}.$$

First compute $$f(0)$$ directly:

$$f(0)=\frac{k}{4}(0)^4-\frac{k}{2}(0)^2+C=C.$$

So the defining condition for membership in $$T$$ is

$$f(x)=C.$$

Substituting the explicit formula for $$f(x)$$ gives

$$\frac{k}{4}\,x^4-\frac{k}{2}\,x^2+C=C.$$

Subtracting $$C$$ from both sides simplifies to

$$\frac{k}{4}\,x^4-\frac{k}{2}\,x^2=0.$$

Because $$k\neq0$$ we may divide through by $$k$$ and then multiply by $$4$$ to clear the fraction:

$$x^4-2x^2=0.$$

We factor completely:

$$x^2\,(x^2-2)=0.$$

This yields two separate equations:

$$x^2=0 \quad\text{or}\quad x^2=2.$$

Solving each gives the real elements of $$T$$:

$$x=0,\quad x=\sqrt2,\quad x=-\sqrt2.$$

Hence

$$T=\{\,0,\;\sqrt2,\;-\sqrt2\,\}.$$

We are asked for the sum of the squares of all elements of $$T$$. Squaring each element we have

$$0^2=0,\qquad(\sqrt2)^2=2,\qquad(-\sqrt2)^2=2.$$

Adding these results:

$$0+2+2=4.$$

Hence, the correct answer is Option A.

We have the function

$$f(x)=\bigl(1-\cos^2x\bigr)\,(\lambda+\sin x),\qquad x\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right),\qquad\lambda\in\mathbb R.$$

Because the Pythagorean identity says $$\sin^2x+\cos^2x=1,$$ we may write

$$1-\cos^2x=\sin^2x.$$

Substituting this into the definition of $$f(x)$$ gives

$$f(x)=\sin^2x\;(\lambda+\sin x).$$

To locate local maxima and minima we set the first derivative equal to zero. First we differentiate. Write

$$g(x)=\sin^2x,\qquad h(x)=\lambda+\sin x,$$

so that $$f(x)=g(x)\,h(x).$$ Using the product rule $$\bigl(g\,h\bigr)'=g'h+gh',$$ and the facts

$$g'(x)=2\sin x\cos x=\sin 2x,\qquad h'(x)=\cos x,$$

we obtain

$$f'(x)=\sin 2x\;(\lambda+\sin x)+\sin^2x\;\cos x.$$

Replacing $$\sin 2x$$ by $$2\sin x\cos x$$ simplifies the expression:

$$f'(x)=2\sin x\cos x\;(\lambda+\sin x)+\sin^2x\cos x.$$

Both terms contain the factor $$\sin x\cos x,$$ so we factor it out:

$$f'(x)=\sin x\cos x\;\bigl[\,2(\lambda+\sin x)+\sin x\,\bigr].$$

Simplifying the bracket further,

$$2(\lambda+\sin x)+\sin x=2\lambda+2\sin x+\sin x=2\lambda+3\sin x.$$

Hence

$$f'(x)=\sin x\cos x\;\bigl(2\lambda+3\sin x\bigr).$$

Critical points arise when $$f'(x)=0,$$ i.e.

$$\sin x=0\quad\text{or}\quad\cos x=0\quad\text{or}\quad2\lambda+3\sin x=0.$$

Inside the open interval $$\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$$ we note

• $$\cos x=0$$ only at $$x=\pm\dfrac{\pi}{2},$$ which are not in the domain, so they are discarded.

• $$\sin x=0$$ gives the critical point $$x_0=0.$$

• $$2\lambda+3\sin x=0$$ yields

$$\sin x=-\dfrac{2\lambda}{3}.$$

This equation has a real solution in the given interval precisely when

$$-1<-\dfrac{2\lambda}{3}<1.$$

Multiplying throughout by $$-\dfrac{3}{2}$$ (remembering to reverse the inequalities) gives

$$-\,\dfrac{3}{2}<\lambda<\dfrac{3}{2}.$$

If $$\lambda=0,$$ the value $$-\dfrac{2\lambda}{3}=0$$ coincides with the already found root $$x_0=0,$$ so we would have only one critical point. To obtain exactly two distinct critical points we therefore require

$$\lambda\in\left(-\dfrac{3}{2},\dfrac{3}{2}\right)\setminus\{0\}.$$

Let $$x_1$$ denote the additional root satisfying $$\sin x_1=-\dfrac{2\lambda}{3}\neq0.$$ The two critical points are thus

$$x_1\quad\text{and}\quad x_0=0.$$

We now examine the nature (maxima or minima) of these points by studying the sign of $$f'(x).$$ The factorisation

$$f'(x)=\bigl(\sin x\bigr)\bigl(\cos x\bigr)\bigl(2\lambda+3\sin x\bigr)$$

makes the sign analysis straightforward because $$\cos x>0$$ throughout the interval. Only the signs of $$\sin x$$ and $$2\lambda+3\sin x$$ matter.

Case 1: $$\lambda>0.$$

Then $$-\dfrac{2\lambda}{3}<0,$$ so $$x_1<0.$$ Moving from left to right along the interval we have

• $$( -\dfrac{\pi}{2},\,x_1):\quad\sin x<0,\;2\lambda+3\sin x<0\;\Longrightarrow\;f'(x)>0;$$
• $$x=x_1:\;f'(x)=0;$$
• $$(x_1,\,0):\quad\sin x<0,\;2\lambda+3\sin x>0\;\Longrightarrow\;f'(x)<0;$$
• $$x=0:\;f'(x)=0;$$
• $$(0,\,\dfrac{\pi}{2}):\quad\sin x>0,\;2\lambda+3\sin x>0\;\Longrightarrow\;f'(x)>0.$$

The derivative changes from positive to negative at $$x_1,$$ giving a local maximum there, and from negative to positive at $$x_0=0,$$ giving a local minimum. Thus we obtain exactly one maxima and one minima.

Case 2: $$\lambda<0.$$

Now $$-\dfrac{2\lambda}{3}>0,$$ so $$x_1>0.$$ The sign table becomes

• $$( -\dfrac{\pi}{2},\,0):\quad\sin x<0,\;2\lambda+3\sin x<0\;\Longrightarrow\;f'(x)>0;$$
• $$x=0:\;f'(x)=0;$$
• $$(0,\,x_1):\quad\sin x>0,\;2\lambda+3\sin x<0\;\Longrightarrow\;f'(x)<0;$$
• $$x=x_1:\;f'(x)=0;$$
• $$(x_1,\,\dfrac{\pi}{2}):\quad\sin x>0,\;2\lambda+3\sin x>0\;\Longrightarrow\;f'(x)>0.$$

Here the derivative switches from positive to negative at $$x_0=0,$$ producing a local maximum, and from negative to positive at $$x_1,$$ producing a local minimum. Again there is exactly one maxima and one minima.

Consequently, every $$\lambda$$ satisfying

$$\boxed{\,-\dfrac{3}{2}<\lambda<\dfrac{3}{2},\;\lambda\neq0\,}$$

gives exactly two distinct critical points, one a maximum and the other a minimum. No other value of $$\lambda$$ works:

• If $$\lambda=\pm\dfrac{3}{2},$$ the second root would correspond to $$\sin x=\mp1,$$ i.e. to $$x=\pm\dfrac{\pi}{2},$$ outside the open interval, leaving only a single critical point.

• If $$\lambda=0,$$ both conditions $$\sin x=0$$ and $$2\lambda+3\sin x=0$$ coincide, again giving only one critical point.

The required set is therefore

$$\left(-\dfrac{3}{2},\dfrac{3}{2}\right)\setminus\{0\}.$$

Hence, the correct answer is Option D.

For applying Lagrange’s Mean Value Theorem we start with its statement: for a function which is continuous on a closed interval $$[a,b]$$ and differentiable on the open interval $$(a,b)$$, there exists at least one point $$c \in (a,b)$$ such that

$$f'(c)=\dfrac{f(b)-f(a)}{b-a}.$$

Here we have $$f(x)=x^{3}-4x^{2}+8x+11$$ defined on $$[0,1]$$, so we take $$a=0$$ and $$b=1$$. The given polynomial is continuous and differentiable everywhere, hence all the conditions are satisfied.

First we compute the values of the function at the endpoints:

$$\begin{aligned} f(1) &= 1^{3}-4(1)^{2}+8(1)+11 \\ &= 1-4+8+11 \\ &= 16, \\ f(0) &= 0^{3}-4(0)^{2}+8(0)+11 \\ &= 11. \end{aligned}$$

Now the average rate of change (the slope of the chord) on $$[0,1]$$ is

$$\dfrac{f(1)-f(0)}{1-0}=\dfrac{16-11}{1}=5.$$

Next we find the derivative of the function:

$$\begin{aligned} f(x)&=x^{3}-4x^{2}+8x+11,\\ f'(x)&=3x^{2}-8x+8. \end{aligned}$$

By the theorem, at the required point $$c$$ we must have

$$f'(c)=5.$$

So we set

$$3c^{2}-8c+8=5.$$

Bringing all terms to one side gives

$$3c^{2}-8c+8-5=0,$$

which simplifies to

$$3c^{2}-8c+3=0.$$

We solve this quadratic equation. The quadratic formula states that for $$Ax^{2}+Bx+C=0$$ the roots are

$$x=\dfrac{-B\pm\sqrt{B^{2}-4AC}}{2A}.$$

Here $$A=3,\;B=-8,\;C=3$$, so

$$\begin{aligned} c &=\dfrac{-(-8)\pm\sqrt{(-8)^{2}-4\cdot3\cdot3}}{2\cdot3}\\ &=\dfrac{8\pm\sqrt{64-36}}{6}\\ &=\dfrac{8\pm\sqrt{28}}{6}. \end{aligned}$$

Because $$\sqrt{28}=2\sqrt{7}$$, this becomes

$$c=\dfrac{8\pm2\sqrt{7}}{6}=\dfrac{4\pm\sqrt{7}}{3}.$$

Thus we obtain two possible values:

$$c_{1}=\dfrac{4+\sqrt{7}}{3},\qquad c_{2}=\dfrac{4-\sqrt{7}}{3}.$$

We now check which of these lies inside the open interval $$(0,1)$$:

$$\begin{aligned} c_{1}&=\dfrac{4+\sqrt{7}}{3}\approx\dfrac{4+2.6458}{3}\approx2.215\; >1,\\ c_{2}&=\dfrac{4-\sqrt{7}}{3}\approx\dfrac{4-2.6458}{3}\approx0.451\; \in(0,1). \end{aligned}$$

Only $$c=\dfrac{4-\sqrt{7}}{3}$$ satisfies $$0<c<1$$, so this is the required point ensured by Lagrange’s Mean Value Theorem.

Hence, the correct answer is Option B.

Let us denote by $$r = 10\;{\rm cm}$$ the fixed radius of the iron ball and by $$y(t)$$ the thickness of the surrounding ice at time $$t$$ minutes. Therefore, the outer radius of the ice sphere is $$R(t)=r+y(t)=10+y(t)$$.

The volume of just the ice layer is obtained by subtracting the iron volume from the total volume of the ice-covered sphere. Using the standard volume formula for a sphere, $$V=\dfrac{4}{3}\pi(\text{radius})^{3}$$, we have

$$ V(t)=\frac{4}{3}\pi\Bigl[R(t)^{3}-r^{3}\Bigr] =\frac{4}{3}\pi\Bigl[(10+y)^{3}-10^{3}\Bigr]. $$

Given data state that the ice is melting at a rate of $$50\;{\rm cm^{3}/min}$$, so the volume is decreasing. Hence $$\dfrac{dV}{dt}=-50\;{\rm cm^{3}/\min}$$ (negative sign because the volume is going down).

To relate this to the change in thickness, we differentiate the expression for $$V(t)$$ with respect to time. First rewrite the inner function: $$R(t)=10+y$$, so $$\dfrac{dR}{dt}=\dfrac{dy}{dt}$$ because the iron ball itself does not change size.

Differentiate: using the chain rule and the fact that $$\dfrac{d}{dt}(R^{3})=3R^{2}\dfrac{dR}{dt}$$, we get

$$ \frac{dV}{dt}= \frac{4}{3}\pi\bigl[3R^{2}\tfrac{dR}{dt}\bigr] =4\pi R^{2}\frac{dR}{dt}. $$

Since $$\dfrac{dR}{dt}=\dfrac{dy}{dt}$$, we can write

$$ \frac{dV}{dt}=4\pi R^{2}\frac{dy}{dt}. $$

We are interested in the instant when the ice thickness is $$y=5\;{\rm cm}$$, so the outer radius is

$$ R=10+5=15\;{\rm cm}. $$

Substituting $$\dfrac{dV}{dt}=-50$$ and $$R=15$$ into the differentiated formula:

$$ -50 = 4\pi (15)^{2}\frac{dy}{dt}. $$

Simplify the numerical factor:

$$ -50 = 4\pi \times 225 \times \frac{dy}{dt} = 900\pi \,\frac{dy}{dt}. $$

Now isolate $$\dfrac{dy}{dt}$$:

$$ \frac{dy}{dt}=\frac{-50}{900\pi} =\frac{-1}{18\pi}\;{\rm cm/min}. $$

The negative sign confirms that the thickness is decreasing. The rate at which the thickness decreases, asked as a positive quantity, is

$$ \bigl|\tfrac{dy}{dt}\bigr|=\frac{1}{18\pi}\;{\rm cm/min}. $$

Hence, the correct answer is Option D.

We are given the function $$f(x)=\log_e\!\left(\dfrac{x^2+\alpha}{7x}\right)$$ on the closed interval $$[3,4]$$ and we are told that Rolle’s theorem holds for some real number $$\alpha$$. For Rolle’s theorem to be applicable, the function must be continuous on the closed interval, differentiable on the open interval and, most importantly, must satisfy $$f(3)=f(4)$$. Continuity and differentiability are already guaranteed by the form of the function, so we begin by imposing the equality of the end-point values.

First we evaluate the function at the two end points:

$$f(3)=\ln\!\left(\dfrac{3^2+\alpha}{7\cdot3}\right)=\ln\!\left(\dfrac{9+\alpha}{21}\right),$$ $$f(4)=\ln\!\left(\dfrac{4^2+\alpha}{7\cdot4}\right)=\ln\!\left(\dfrac{16+\alpha}{28}\right).$$

Rolle’s condition $$f(3)=f(4)$$ therefore becomes

$$\ln\!\left(\dfrac{9+\alpha}{21}\right)=\ln\!\left(\dfrac{16+\alpha}{28}\right).$$

Since the natural logarithm is one-to-one, the arguments must be equal:

$$\dfrac{9+\alpha}{21}=\dfrac{16+\alpha}{28}.$$

Cross-multiplying gives

$$28(9+\alpha)=21(16+\alpha).$$

Expanding both sides we get

$$252+28\alpha=336+21\alpha.$$

Subtracting $$21\alpha$$ and $$252$$ respectively, we obtain

$$7\alpha=84,$$ $$\alpha=12.$$

Now we know the exact function relevant to the problem:

$$f(x)=\ln\!\left(\dfrac{x^2+12}{7x}\right)=\ln(x^2+12)-\ln7-\ln x.$$

Next, Rolle’s theorem states that there exists some $$c\in(3,4)$$ with $$f'(c)=0$$. So we must first compute the derivative $$f'(x)$$.

Writing the logarithm as a difference, we differentiate term by term:

$$f(x)=\ln(x^2+12)-\ln7-\ln x,$$ $$f'(x)=\dfrac{2x}{x^2+12}-\dfrac{1}{x}.$$

We now set $$f'(c)=0$$:

$$\dfrac{2c}{c^2+12}-\dfrac{1}{c}=0.$$

Bringing terms to a common denominator and cross-multiplying yields

$$\dfrac{2c^2}{c^2+12}=1,$$ $$2c^2=c^2+12,$$ $$c^2=12,$$ $$c=2\sqrt3.$$

The positive root is taken because $$c$$ must lie in the interval $$[3,4]$$, and indeed $$2\sqrt3\approx3.464$$ satisfies $$3<2\sqrt3<4$$.

The problem asks for $$f''(c)$$, so we next compute the second derivative $$f''(x)$$. To do that we differentiate $$f'(x)$$ once more, taking each term separately.

We start with $$f'(x)=\dfrac{2x}{x^2+12}-\dfrac{1}{x}.$$ The derivative of the first fraction is found through the quotient rule. If we let $$g(x)=2x$$ and $$h(x)=x^2+12$$, then the quotient rule, $$\left(\dfrac{g}{h}\right)'=\dfrac{g'h-gh'}{h^2}$$, gives

$$\left(\dfrac{2x}{x^2+12}\right)'=\dfrac{2(x^2+12)-2x\cdot2x}{(x^2+12)^2}=\dfrac{2x^2+24-4x^2}{(x^2+12)^2}=\dfrac{-2x^2+24}{(x^2+12)^2}.$$

The derivative of $$-\dfrac1x$$ is $$+\dfrac1{x^2}$$, because $$\dfrac{d}{dx}(-x^{-1})=+x^{-2}$$.

Adding these two parts, we obtain the full second derivative:

$$f''(x)=\dfrac{-2x^2+24}{(x^2+12)^2}+\dfrac1{x^2}.$$

We now evaluate this at $$x=c=2\sqrt3$$. For this value we have

$$c^2=(2\sqrt3)^2=12,$$ $$c^2+12=12+12=24.$$

Substituting into the first fraction numerator we get

$$-2c^2+24=-2(12)+24=-24+24=0,$$ so the entire first fraction becomes

$$\dfrac{0}{(24)^2}=0.$$

The second term gives $$\dfrac1{c^2}=\dfrac1{12}.$$

Therefore

$$f''(c)=0+\dfrac1{12}=\dfrac1{12}.$$

Hence, the correct answer is Option B.

We consider a cube whose edge length (side) at any instant is denoted by $$a \;{\rm cm}$$. Two basic geometrical relations for a cube are needed.

First, the surface area $$S$$ of a cube is given by the formula

$$S \;=\; 6a^{2}.$$

Second, the volume $$V$$ of a cube is given by the formula

$$V \;=\; a^{3}.$$

Both $$S$$ and $$V$$ are changing with time because the edge length $$a$$ itself is changing with time. Let $$\dfrac{dS}{dt}$$ and $$\dfrac{dV}{dt}$$ denote their respective rates of change, and let $$\dfrac{da}{dt}$$ be the rate at which the edge length changes.

We are told that the surface area is increasing at a constant rate of $$3.6\;{\rm cm^{2}/s}$$, so

$$\dfrac{dS}{dt} \;=\; 3.6.$$

Our target is $$\dfrac{dV}{dt}$$ when $$a = 10\;{\rm cm}$$.

We begin by differentiating the surface‐area formula with respect to time $$t$$. Since $$S = 6a^{2}$$, we have

$$\frac{dS}{dt} \;=\; \frac{d}{dt}\bigl(6a^{2}\bigr) \;=\; 6 \cdot 2a \cdot \frac{da}{dt} \;=\; 12a\,\frac{da}{dt}.$$

Now we solve this relation for $$\dfrac{da}{dt}$$:

$$\frac{da}{dt} \;=\; \frac{1}{12a}\,\frac{dS}{dt}.$$

Substituting the given value $$\dfrac{dS}{dt}=3.6$$ and the specific edge length $$a = 10$$, we obtain

$$\frac{da}{dt} \;=\; \frac{1}{12 \times 10}\,(3.6) \;=\; \frac{3.6}{120} \;=\; 0.03\;{\rm cm/s}.$$

Next, we differentiate the volume formula. From $$V = a^{3}$$,

$$\frac{dV}{dt} \;=\; \frac{d}{dt}\bigl(a^{3}\bigr) \;=\; 3a^{2}\,\frac{da}{dt}.$$

We already know $$a = 10$$ and $$\dfrac{da}{dt} = 0.03$$, so substitute these values:

$$\frac{dV}{dt} \;=\; 3 \times (10)^{2} \times 0.03 \;=\; 3 \times 100 \times 0.03 \;=\; 300 \times 0.03 \;=\; 9\;{\rm cm^{3}/s}.$$

Therefore, the volume of the cube is increasing at the rate of $$9\;{\rm cm^{3}/s}$$ when the edge length is $$10\;{\rm cm}$$.

Hence, the correct answer is Option D.

We are given the curve $$y = x + \sin y$$ and consider a point $$(a,b)$$ on it. To find the slope of the tangent, we differentiate implicitly with respect to $$x$$.

First write the equation again so that the differentiation step is clear:

$$y = x + \sin y.$$

Differentiating both sides with respect to $$x$$, we remember that $$\dfrac{d}{dx}(y)=\dfrac{dy}{dx}$$ and that $$\dfrac{d}{dx}(\sin y)=\cos y \dfrac{dy}{dx}$$ because $$y$$ itself depends on $$x$$. Hence

$$\frac{dy}{dx} = 1 + \cos y \,\frac{dy}{dx}.$$

Now we collect the $$\dfrac{dy}{dx}$$ terms on the left:

$$\frac{dy}{dx} - \cos y \,\frac{dy}{dx} = 1.$$

Factor out $$\dfrac{dy}{dx}:$$

$$\left(1 - \cos y\right)\frac{dy}{dx} = 1.$$

So, solving for the derivative, we obtain

$$\frac{dy}{dx} = \frac{1}{1 - \cos y}.$$

At the specific point $$(a,b)$$ on the curve, the slope of the tangent (call it $$m_T$$) is therefore

$$m_T = \frac{1}{1 - \cos b}.$$

The problem states that this tangent is parallel to the line joining the two points $$\bigl(0,\tfrac32\bigr)$$ and $$\bigl(\tfrac12,2\bigr).$$ We first compute the slope of that line. The two-point slope formula is

$$m = \frac{y_2 - y_1}{x_2 - x_1}.$$

Substituting $$\bigl(x_1,y_1\bigr)=(0,\tfrac32)$$ and $$\bigl(x_2,y_2\bigr)=\bigl(\tfrac12,2\bigr)$$, we get

$$m = \frac{2 - \tfrac32}{\tfrac12 - 0} = \frac{\tfrac12}{\tfrac12} = 1.$$

Because the two lines are parallel, their slopes are equal. Thus

$$m_T = 1.$$

So we set

$$\frac{1}{1 - \cos b} = 1.$$

Cross-multiplying gives

$$1 = 1 - \cos b,$$ $$\cos b = 0.$$

The solutions of $$\cos b = 0$$ are

$$b = \frac{\pi}{2} + k\pi,\quad k \in \mathbb{Z}.$$

Next, we must relate $$a$$ and $$b$$ using the original curve equation $$y = x + \sin y.$$ At the point $$(a,b)$$ we substitute:

$$b = a + \sin b.$$

Rearrange to express $$a$$ in terms of $$b$$:

$$a = b - \sin b.$$

Because $$b = \dfrac{\pi}{2} + k\pi,$$ we note that $$\sin b = \sin\!\Bigl(\dfrac{\pi}{2}+k\pi\Bigr) = (-1)^k \cdot 1.$$ Hence $$\sin b = \pm 1,$$ and therefore

$$a = b \mp 1.$$

Taking the absolute difference, we have

$$|b - a| = |\,\sin b\,| = 1.$$

This precisely matches Option B. No other listed option is guaranteed for every integer $$k$$.

Hence, the correct answer is Option B.

Let $$f(x)$$ be a polynomial of degree 5. Since $$\lim_{x \to 0}\left(2 + \frac{f(x)}{x^3}\right) = 4$$, the limit $$\lim_{x \to 0} \frac{f(x)}{x^3}$$ must exist and equal 2.

For $$\frac{f(x)}{x^3}$$ to have a finite limit as $$x \to 0$$, $$f(x)$$ must have no constant term, no $$x$$ term, and no $$x^2$$ term. So:

$$f(x) = ax^5 + bx^4 + cx^3$$

Then $$\frac{f(x)}{x^3} = ax^2 + bx + c$$, and $$\lim_{x \to 0} \frac{f(x)}{x^3} = c = 2$$.

So $$f(x) = ax^5 + bx^4 + 2x^3$$.

Since $$x = \pm 1$$ are critical points, we need $$f'(\pm 1) = 0$$.

$$f'(x) = 5ax^4 + 4bx^3 + 6x^2$$

$$f'(1) = 5a + 4b + 6 = 0$$ $$-(1)$$

$$f'(-1) = 5a - 4b + 6 = 0$$ $$-(2)$$

Adding $$(1)$$ and $$(2)$$: $$10a + 12 = 0$$, so $$a = -\frac{6}{5}$$.

Subtracting $$(2)$$ from $$(1)$$: $$8b = 0$$, so $$b = 0$$.

Therefore: $$f(x) = -\frac{6}{5}x^5 + 2x^3$$

Now we verify each option.

Option A: $$f$$ is an odd function.

$$f(-x) = -\frac{6}{5}(-x)^5 + 2(-x)^3 = \frac{6}{5}x^5 - 2x^3 = -f(x)$$

So $$f$$ is indeed an odd function. Option A is true.

Option B: $$f(1) - 4f(-1) = 4$$.

$$f(1) = -\frac{6}{5} + 2 = \frac{4}{5}$$

$$f(-1) = \frac{6}{5} - 2 = -\frac{4}{5}$$

$$f(1) - 4f(-1) = \frac{4}{5} - 4\left(-\frac{4}{5}\right) = \frac{4}{5} + \frac{16}{5} = \frac{20}{5} = 4$$

Option B is true.

Option C: $$x = 1$$ is a point of local minimum and $$x = -1$$ is a point of local maximum.

We compute the second derivative:

$$f''(x) = -24x^3 + 12x$$

$$f''(1) = -24 + 12 = -12 < 0$$, so $$x = 1$$ is a local maximum (not minimum).

$$f''(-1) = 24 - 12 = 12 > 0$$, so $$x = -1$$ is a local minimum (not maximum).

Option C states the exact opposite of what is true. Option C is not true.

Option D: $$x = 1$$ is a point of local maxima of $$f$$.

Since $$f''(1) = -12 < 0$$, $$x = 1$$ is indeed a local maximum. Option D is true.

The statement that is NOT true is Option C.

We have a function $$f:[-7,0]\to\mathbb R$$ which is continuous on the closed interval $$[-7,0]$$ and differentiable on the open interval $$(-7,0)$$. The values already known are $$f(-7)=-3$$ and, for every $$x\in(-7,0)$$, the derivative satisfies $$f'(x)\le 2$$.

To relate the values of the function at different points we use the Mean Value Theorem (MVT). The MVT states that for any two points $$a,b$$ in the interval with $$a<b$$, there exists a point $$c\in(a,b)$$ such that

$$f(b)-f(a)=f'(c)\,(b-a).$$

Because $$f'(c)\le 2$$, the difference $$f(b)-f(a)$$ cannot exceed $$2(b-a)$$. Expressed as an inequality,

$$f(b)\le f(a)+2\,(b-a).$$

We now apply this inequality twice, once for the pair $$(-7,-1)$$ and once for the pair $$(-7,0)$$.

First take $$a=-7,\;b=-1$$. Then $$b-a=(-1)-(-7)=6$$, so

$$f(-1)\le f(-7)+2\cdot 6 =-3+12 =9.$$ Thus

$$f(-1)\le 9.$$

Next take $$a=-7,\;b=0$$. Now $$b-a=0-(-7)=7$$, hence

$$f(0)\le f(-7)+2\cdot 7 =-3+14 =11.$$ So

$$f(0)\le 11.$$

Adding the two individual inequalities gives an upper bound for the required sum:

$$f(-1)+f(0)\le 9+11=20.$$ Hence, whatever the admissible function,

$$f(-1)+f(0)\le 20.$$

We must also decide whether there is a finite lower bound. Observe that the derivative condition $$f'(x)\le 2$$ places no restriction on how negative the derivative may be; it can be any negative number. Therefore the function is free to decrease as steeply as we like.

To see this concretely, fix a very large positive number $$N$$ and construct a simple piecewise-linear example:

For $$x\in[-7,-1]$$ set $$f(x)=-3-N\,(x+7).$$ Here the constant slope is $$f'(x)=-N$$, which respects the inequality $$f'(x)\le 2$$ because $$-N<2$$ for every positive $$N$$.

At $$x=-1$$ the value is

$$f(-1)=-3-N\,(6)=-3-6N.$$

On the short interval $$[-1,0]$$, let the function rise with the maximum permitted slope $$2$$: $$f(x)=f(-1)+2\,(x+1),\qquad x\in[-1,0].$$ This keeps the derivative equal to $$2$$ and ensures differentiability at $$x=-1$$.

Then at $$x=0$$ we obtain

$$f(0)=f(-1)+2\,(0+1) =(-3-6N)+2 =-1-6N.$$

The sum of the two required values is therefore

$$f(-1)+f(0)=(-3-6N)+(-1-6N)=-4-12N.$$

As $$N\to\infty$$ the number $$-4-12N$$ tends to $$-\infty$$. So by choosing a sufficiently large $$N$$, the sum $$f(-1)+f(0)$$ can be made arbitrarily negative. Consequently, there is no finite lower bound; the set of all possible values extends indefinitely towards $$-\infty$$.

Combining the results, we have established that

$$-\infty<f(-1)+f(0)\le 20.$$

Thus the entire range of admissible sums is the interval $$(-\infty,\,20]$$.

Hence, the correct answer is Option A.

We have the real-valued function

$$f(x)=\,(3x-7)\,x^{\frac23},\qquad x\in\mathbb R.$$

To find where the function is increasing, we must find the sign of its first derivative. A function is increasing wherever $$f'(x)>0.$$ We first compute the derivative.

By the Product Rule, $$\dfrac{d}{dx}[u\;v]=u'\,v+u\,v',$$ with $$u=3x-7,\qquad v=x^{\frac23}.$$

We differentiate each part:

$$u'= \dfrac{d}{dx}(3x-7)=3,$$ $$v'= \dfrac{d}{dx}\!\left(x^{\frac23}\right)=\frac23\,x^{\frac23-1}=\frac23\,x^{-\frac13}.$$

Substituting these into the Product Rule, we obtain

$$ \begin{aligned} f'(x) & = u'\,v+u\,v' \\ & = 3\,x^{\frac23} \;+\;(3x-7)\left(\frac23\,x^{-\frac13}\right). \end{aligned} $$

Now we simplify. Notice that $$x^{\frac23}=x^{-\frac13}\,x,$$ so we can factor out the common $$x^{-\frac13}$$ term:

$$ \begin{aligned} f'(x) & = x^{-\frac13}\!\left[\,3x + \frac23\,(3x-7)\right]. \end{aligned} $$

We next combine the terms inside the brackets. First write everything with a common denominator 3:

$$3x = \frac{9x}{3},\qquad \frac23\,(3x-7)=\frac{2(3x-7)}{3}.$$

Hence,

$$ \begin{aligned} 3x + \frac23(3x-7) &= \frac{9x}{3}+\frac{2(3x-7)}{3}\\[2pt] &= \frac{9x+6x-14}{3}\\[2pt] &= \frac{15x-14}{3}. \end{aligned} $$

Therefore, the derivative becomes

$$ f'(x)=x^{-\frac13}\,\frac{15x-14}{3}. $$

The constant denominator 3 is positive, so the sign of $$f'(x)$$ depends only on the two factors

$$x^{-\frac13}\quad\text{and}\quad(15x-14).$$

Recall that $$x^{-\frac13}=1/x^{\frac13}.$$ The real cube root preserves the sign of $$x,$$ so

• For $$x>0,$$ we have $$x^{-\frac13}>0.$$
• For $$x<0,$$ we have $$x^{-\frac13}<0.$$

The other factor, $$15x-14,$$ changes sign at $$x=\frac{14}{15}.$$ Its sign is

• Positive if $$x>\frac{14}{15},$$
• Negative if $$x<\frac{14}{15}.$$

We consider the two regions of $$x$$ separately.

1. Region $$x>0$$. Here $$x^{-\frac13}>0.$$ So $$f'(x)>0$$ when the second factor is positive, i.e.

$$15x-14>0\quad\Longrightarrow\quad x>\frac{14}{15}.$$

Hence, on the positive side the function is increasing for $$x\in\left(\frac{14}{15},\infty\right).$$

2. Region $$x<0$$. Here $$x^{-\frac13}<0.$$ For the product to be positive we now need $$15x-14<0,$$ which is automatically true for every negative $$x.$$ Thus,

$$f'(x)>0\quad\text{for all }x\in(-\infty,0).$$

The derivative is undefined at $$x=0,$$ but that single point does not affect intervals of monotonicity. Collecting the two increasing portions, we conclude that

$$f(x)\text{ is increasing on }(-\infty,0)\cup\left(\frac{14}{15},\infty\right).$$

This set of intervals matches Option A.

Hence, the correct answer is Option A.