JEE Applications of Derivatives PYQs with Video Solutions PDF

Consider a quadratic equation $$ax^2 + bx + c = 0$$, where $$2a + 3b + 6c = 0$$ and let $$g(x) = a\frac{x^3}{3} + b\frac{x^2}{2} + cx$$. Statement 1: The quadratic equation has at least one root in the interval $$(0,1)$$. Statement 2: The Rolle's theorem is applicable to function $$g(x)$$ on the interval $$[0,1]$$.

If the function $$f(x) = \left(\frac{1}{x}\right)^{2x}; x > 0$$ attains the maximum value at $$x = \frac{1}{e}$$ then :

Let $$g(x) = 3f\left(\frac{x}{3}\right) + f(3 - x)$$ and $$f''(x) > 0$$ for all $$x \in (0, 3)$$. If g is decreasing in $$(0, \alpha)$$ and increasing in $$(\alpha, 3)$$, then $$8\alpha$$ is

The local maximum value of the function, $$f(x) = \left(\frac{2}{x}\right)^{x^2}$$, $$x \gt 0$$, is:

The function $$f(x) = 2x + 3x^{\frac{2}{3}}, x \in R$$, has

Let $$x = -1$$ and $$x = 2$$ be the critical points of the function $$f(x) = x^3 + ax^2 + b \log_e|x| + 1, x \neq 0$$. Let m and M respectively be the absolute minimum and the absolute maximum values of f in the interval $$\left[-2, -\frac{1}{2}\right]$$. Then $$|M + m|$$ is equal to :
(Take $$\log_{e}2 = 0.7$$)

If the absolute maximum value of the function $$f(x) = (x^2 - 2x + 7)e^{(4x^3 - 12x^2 - 180x + 31)}$$ in the interval $$[-3, 0]$$ is $$f(\alpha)$$, then

The sum of the maximum and minimum values of the function $$f(x) = |5x - 7| + [x^2 + 2x]$$ in the interval $$\left[\dfrac{5}{4}, 2\right]$$, where $$[t]$$ is the greatest integer $$\le t$$, is ______.

A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is $$k$$ (meter), then $$\left(\frac{4}{\pi} + 1\right)k$$ is equal to _________

Let $$f(x)$$ be a cubic polynomial with $$f(1) = -10$$, $$f(-1) = 6$$, and has a local minima at $$x = 1$$, and $$f'(x)$$ has a local minima at $$x = -1$$. Then $$f(3)$$ is equal to _________.

If the curves $$x = y^4$$ and $$xy = k$$ cut at right angles, then $$(4k)^6$$ is equal to ______.

The triangle of maximum area that can be inscribed in a given circle of radius 'r' is:

The shortest distance between the line $$x - y = 1$$ and the curve $$x^2 = 2y$$ is:

Let $$(2,3)$$  be the largest open interval in which the function $$f(x)=2\log_e(x-2)-x^2+ax+1$$ is strictly increasing and  $$(b,c)$$  be the largest open interval in which the function $$g(x)=(x-1)^3(x+2-a)^2$$ is strictly decreasing. Then  $$100(a+b-c)$$  is equal to:

The maximum area of a triangle whose one vertex is at $$(0, 0)$$ and the other two vertices lie on the curve $$y = -2x^2 + 54$$ at points $$(x, y)$$ and $$(-x, y)$$ where $$y > 0$$ is :

Water is being filled at the rate of $$1$$ cm$$^3$$ sec$$^{-1}$$ in a right circular conical vessel (vertex downwards) of height $$35$$ cm and diameter $$14$$ cm. When the height of the water level is $$10$$ cm, the rate (in cm$$^2$$ sec$$^{-1}$$) at which the wet conical surface area of the vessel increases is

Consider a cuboid of sides $$2x, 4x$$ and $$5x$$ and a closed hemisphere of radius $$r$$. If the sum of their surface areas is constant $$k$$, then the ratio $$x : r$$, for which the sum of their volumes is maximum, is

Let $$f$$ and $$g$$ be twice differentiable even functions on $$(-2, 2)$$ such that $$f\left(\frac{1}{4}\right) = 0$$, $$f\left(\frac{1}{2}\right) = 0$$, $$f(1) = 1$$ and $$g\left(\frac{3}{4}\right) = 0$$, $$g(1) = 2$$. Then, the minimum number of solutions of $$f(x)g''(x) + f'(x)g'(x) = 0$$ in $$(-2, 2)$$ is equal to ______.

Let the area enclosed by the $$x$$-axis, and the tangent and normal drawn to the curve $$4x^3 - 3xy^2 + 6x^2 - 5xy - 8y^2 + 9x + 14 = 0$$ at the point $$(-2, 3)$$ be $$A$$. Then $$8A$$ is equal to ______.

Let $$f(x) = \begin{cases} x^3 - x^2 + 10x - 7, & x \le 1 \\ -2x + \log_2(b^2 - 4), & x > 1 \end{cases}$$. Then the set of all values of $$b$$, for which $$f(x)$$ has maximum value at $$x = 1$$, is:

The sum of the absolute maximum and absolute minimum values of the function $$f(x) = \tan^{-1}(\sin x - \cos x)$$ in the interval $$[0, \pi]$$ is

Let $$f(x) = 3(x^2 - 2)^3 + 4$$, $$x \in \mathbb{R}$$. Then which of the following statements are true?
P: $$x = 0$$ is a point of local minima of f
Q: $$x = \sqrt{2}$$ is a point of inflection of f
R: $$f'$$ is increasing for $$x > \sqrt{2}$$

Let $$f(x)$$ be a polynomial of degree 6 in $$x$$, in which the coefficient of $$x^6$$ is unity and it has extrema at $$x = -1$$ and $$x = 1$$. If $$\lim_{x \to 0} \frac{f(x)}{x^3} = 1$$, then $$5 \cdot f(2)$$ is equal to ______

If $$R$$ is the least value of $$a$$ such that the function $$f(x) = x^2 + ax + 1$$ is increasing on $$[1, 2]$$ and $$S$$ is the greatest value of $$a$$ such that the function $$f(x) = x^2 + ax + 1$$ is decreasing on $$[1, 2]$$, then the value of $$|R - S|$$ is _________.

The maximum value of $$z$$ in the following equation $$z = 6xy + y^2$$, where $$3x + 4y \leq 100$$ and $$4x + 3y \leq 75$$ for $$x \geq 0$$ and $$y \geq 0$$ is ________.

Let $$f : R \to R$$ be defined as $$f(x) = \begin{cases} -\frac{4}{3}x^3 + 2x^2 + 3x, & x > 0 \\ 3xe^x, & x \le 0 \end{cases}$$
Then $$f$$ is increasing function in the interval

Let $$a$$ and $$b$$ respectively be the points of local maximum and local minimum of the function $$f(x) = 2x^3 - 3x^2 - 12x$$. If $$A$$ is the total area of the region bounded by $$y = f(x)$$, the $$x$$-axis and the lines $$x = a$$ and $$x = b$$, then 4A is equal to _________

The function, $$f(x) = (3x - 7)x^{\frac{2}{3}}$$, $$x \in R$$, is increasing for all $$x$$ lying in:

Let $$f : (-1, \infty) \to R$$ be defined by $$f(0) = 1$$ and $$f(x) = \frac{1}{x}\log_e(1 + x)$$, $$x \ne 0$$. Then the function $$f$$:

If the tangent to the curve $$y = f(x) = x\log_e x$$, $$(x > 0)$$ at a point $$(c, f(c))$$ is parallel to the line-segment joining the points $$(1, 0)$$ and $$(e, e)$$, then $$c$$ is equal to: