{"id":26074,"date":"2019-03-14T18:44:52","date_gmt":"2019-03-14T13:14:52","guid":{"rendered":"https:\/\/cracku.in\/blog\/?p=26074"},"modified":"2019-03-14T18:44:52","modified_gmt":"2019-03-14T13:14:52","slug":"maxima-and-minima-questions-for-cat","status":"publish","type":"post","link":"https:\/\/cracku.in\/blog\/maxima-and-minima-questions-for-cat\/","title":{"rendered":"Maxima and Minima Questions for CAT"},"content":{"rendered":"

Maxima and Minima Questions for CAT:<\/strong><\/span><\/h2>\n

Download important CAT Maxima and Minima Questions PDF based on previous asked questions in CAT and other MBA exams. Top 25 Maxima and Minima functions questions for CAT quantitative aptitude.<\/p>\n

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Question 1:\u00a0<\/b>If y is a real number, what is the difference in the maximum and minimum values obtained by $\\frac{y+5}{y^2+5y+25}$ ?<\/p>\n

a)\u00a02\/15<\/p>\n

b)\u00a04\/15<\/p>\n

c)\u00a01\/5<\/p>\n

d)\u00a01\/15<\/p>\n

Question 2:\u00a0<\/b>What is the maximum value of g(x) = 22 – |x+8| and for what value of x is this reached?<\/p>\n

a)\u00a022, 8<\/p>\n

b)\u00a014, 0<\/p>\n

c)\u00a014, -8<\/p>\n

d)\u00a0None of these<\/p>\n

Question 3:\u00a0<\/b>If 2 $\\leq$ x $\\leq$ 3 and 4 $\\leq$ y $\\leq$ 5, what are the minimum and maximum values of $\\frac{y+x}{y+2x}$<\/p>\n

a)\u00a09\/10, 7\/8<\/p>\n

b)\u00a010\/11, 8\/9<\/p>\n

c)\u00a07\/10, 7\/9<\/p>\n

d)\u00a0None of these<\/p>\n

Question 4:\u00a0<\/b>If $x^{2} – 5x + 4 <= 0$ and $y^{2} -6y+5 <= 0$, what are the maximum and minimum values of $\\frac{y+7x}{y+4x}$<\/p>\n

a)\u00a029\/17 , 12\/9<\/p>\n

b)\u00a023\/13, 11\/8<\/p>\n

c)\u00a025\/11, 14\/11<\/p>\n

d)\u00a0None of these<\/p>\n

Question 5:\u00a0<\/b>If X and Y are positive real numbers and 3X+4Y = 14, what is the maximum value of $X^{3}*Y^4$ ?<\/p>\n

a)\u00a064<\/p>\n

b)\u00a0128<\/p>\n

c)\u00a02187<\/p>\n

d)\u00a0None of these<\/p>\n

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Question 6:\u00a0<\/b>What is the maximum value of g(x) = |34-x| – |x + 11|?<\/p>\n

a)\u00a034<\/p>\n

b)\u00a045<\/p>\n

c)\u00a023<\/p>\n

d)\u00a0None of these<\/p>\n

Question 7:\u00a0<\/b>If X and Y are positive and X+Y = 10, what is the maximum value of $X^3*{Y^2}$ ?<\/p>\n

a)\u00a03125<\/p>\n

b)\u00a03456<\/p>\n

c)\u00a03600<\/p>\n

d)\u00a0None of these<\/p>\n

Question 8:\u00a0<\/b>If f(x) = min(3x-4, 11-7x). What is the maximum value of f(x)?<\/p>\n

a)\u00a03\/2<\/p>\n

b)\u00a0-3\/2<\/p>\n

c)\u00a01\/2<\/p>\n

d)\u00a0Infinite<\/p>\n

Question 9:\u00a0<\/b>For how many real values does the function $F(X) = 28 – |X^2+6|$ attain its maximum value<\/p>\n

a)\u00a01<\/p>\n

b)\u00a02<\/p>\n

c)\u00a03<\/p>\n

d)\u00a0None of these<\/p>\n

Question 10:\u00a0<\/b>If ‘a’ and ‘b’ are two whole numbers such that |a-2|+|b-3| = 4, what is the maximum value of a*b ?<\/p>\n

a)\u00a024<\/p>\n

b)\u00a018<\/p>\n

c)\u00a014<\/p>\n

d)\u00a020<\/p>\n

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Question 11:\u00a0<\/b>If X,Y and Z are real numbers such that $-6 \\leq X \\leq -2$ and $-4 \\leq Y \\leq 4$ and $3 \\leq Z \\leq 7$. If $W=\\frac{Y}{XZ}$, which of the following is necessarily true?<\/p>\n

a)\u00a0$0 \\leq W \\leq 2$<\/p>\n

b)\u00a0$-1\/2 \\leq W \\leq 1\/2$<\/p>\n

c)\u00a0$-2\/3 \\leq W \\leq 1\/3$<\/p>\n

d)\u00a0$-2\/3 \\leq W \\leq 2\/3$<\/p>\n

Question 12:\u00a0<\/b>Two natural numbers, X and Y, satisfy the equation 2X+Y = 33. Find the maximum value of XY?<\/p>\n

a)\u00a0135<\/p>\n

b)\u00a0140<\/p>\n

c)\u00a0136<\/p>\n

d)\u00a0144<\/p>\n

Question 13:\u00a0<\/b>Two natural numbers, A and B, satisfy the equation 3A+2B = 64. Find the maximum value of ${A*B}$?<\/p>\n

a)\u00a0160<\/p>\n

b)\u00a0162<\/p>\n

c)\u00a0170<\/p>\n

d)\u00a0180<\/p>\n

Question 14:\u00a0<\/b>If a,b and c are three whole numbers such that |a-1|+|b-2|+|c-3|=3, what is the maximum value of (a+1)(b+2)(c+3)?<\/p>\n

a)\u00a0120<\/p>\n

b)\u00a0144<\/p>\n

c)\u00a0112<\/p>\n

d)\u00a0105<\/p>\n

Question 15:\u00a0<\/b>If a,b and c are positive real numbers, what is the minimum value of a*(1\/b+1\/c)+b*(1\/c+1\/a)+c*(1\/a+1\/b)<\/p>\n

a)\u00a09<\/p>\n

b)\u00a06<\/p>\n

c)\u00a03<\/p>\n

d)\u00a027<\/p>\n

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Question 16:\u00a0<\/b>If $x^2 – 5x + 4 \\leq 0$ and $y^2 -6y +5 \\leq 0$, then what are the maximum and minimum values of $(y+7x)\/(y+4x)$.<\/p>\n

a)\u00a029\/17, 12\/9<\/p>\n

b)\u00a023\/13, 11\/8<\/p>\n

c)\u00a025\/11, 14\/11<\/p>\n

d)\u00a029\/11, 11\/9<\/p>\n

e)\u00a0None of these<\/p>\n

Question 17:\u00a0<\/b>Let a, b, c, d be four integers such that a+b+c+d = 4m+1 where m is a positive integer. Given m, which one of the following is necessarily true?<\/p>\n

a)\u00a0The minimum possible value of $a^2 + b^2 + c^2 + d^2$ is $4m^2-2m+1$<\/p>\n

b)\u00a0The minimum possible value of $a^2 + b^2 + c^2 + d^2$ is $4m^2+2m+1$<\/p>\n

c)\u00a0The maximum possible value of $a^2 + b^2 + c^2 + d^2$ is $4m^2-2m+1$<\/p>\n

d)\u00a0The maximum possible value of $a^2 + b^2 + c^2 + d^2$ is $4m^2+2m+1$<\/p>\n

Question 18:\u00a0<\/b>If x, y, z are distinct positive real numbers the $(x^2(y+z) + y^2(x+z) + z^2(x+y))\/xyz$ would be<\/p>\n

a)\u00a0greater than 4<\/p>\n

b)\u00a0greater than 5<\/p>\n

c)\u00a0greater than 6<\/p>\n

d)\u00a0None of the above<\/p>\n

Question 19:\u00a0<\/b>If a, b, c and d are four positive real numbers such that abcd = 1, what is the minimum value of (1 + a)(1+b)(1+c)(1+d)?<\/p>\n

a)\u00a04<\/p>\n

b)\u00a01<\/p>\n

c)\u00a016<\/p>\n

d)\u00a018<\/p>\n

Question 20:\u00a0<\/b>Let x and y be two positive numbers such that $x + y = 1.$ Then the minimum value of $(x+\\frac{1}{x})^2+(y+\\frac{1}{y})^2$ is<\/p>\n

a)\u00a012<\/p>\n

b)\u00a020<\/p>\n

c)\u00a012.5<\/p>\n

d)\u00a013.3<\/p>\n

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Question 21:\u00a0<\/b>If a, b, c and d are four positive real numbers such that abcd = 1, what is the minimum value of (1 + a)(1+b)(1+c)(1+d)?[CAT 2001]<\/p>\n

a)\u00a04<\/p>\n

b)\u00a01<\/p>\n

c)\u00a016<\/p>\n

d)\u00a018<\/p>\n

Question 22:\u00a0<\/b>Let x and y be two positive numbers such that $x + y = 1.$Then the minimum value of $(x+\\frac{1}{x})^2+(y+\\frac{1}{y})^2$ is[CAT 2001]<\/p>\n

a)\u00a012<\/p>\n

b)\u00a020<\/p>\n

c)\u00a012.5<\/p>\n

d)\u00a013.3<\/p>\n

Question 23:\u00a0<\/b>If a, b, c, d are positive numbers, what is the minimum value of (a+b+c+d)*(1\/a+1\/b+1\/c+1\/d)?<\/p>\n

a)\u00a09<\/p>\n

b)\u00a016<\/p>\n

c)\u00a025<\/p>\n

d)\u00a01<\/p>\n

Question 24:\u00a0<\/b>If | r – 6 | = 11 and | 2q – 12 | = 8, what is the minimum possible value of q \/ r?<\/p>\n

a)\u00a0-2\/5<\/p>\n

b)\u00a02\/17<\/p>\n

c)\u00a010\/17<\/p>\n

d)\u00a0None of these<\/p>\n

Question 25:\u00a0<\/b>If ‘$p$’ and ‘$q$’ are two positive rational numbers such that $p+q=1$, which of the following statements is true?<\/p>\n

a)\u00a0The minimum value of $p^{\\frac{1}{4}}+q^{\\frac{1}{4}}$ is $2^\\frac{1}{2}$<\/p>\n

b)\u00a0The maximum value of $p^{\\frac{1}{4}}+q^{\\frac{1}{4}}$ is $2^\\frac{1}{2}$<\/p>\n

c)\u00a0The maximum value of $p^{\\frac{1}{3}}+q^{\\frac{1}{3}}$ is $2^\\frac{2}{3}$<\/p>\n

d)\u00a0The minimum value of $p^{\\frac{1}{3}}+q^{\\frac{1}{3}}$ is $2^\\frac{2}{3}$<\/p>\n

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Answers & Solutions:<\/strong><\/span><\/p>\n

1)\u00a0Answer\u00a0(B)<\/strong><\/p>\n

Let $\\frac{y+5}{y^2+5y+25} = k$<\/p>\n

So, $ky^2+5ky+25k-y-5=0$<\/p>\n

For the above quadriatic equation to have real roots, its discriminant should be greater than or equal to zero.<\/p>\n

$(5k-1)^2 \\geq 4*k*5*(5k-1)$<\/p>\n

$(15k+1)(5k-1)\\leq 0 $<\/p>\n

So $-1\/15 \\leq k \\leq 1\/5$<\/p>\n

So, the difference in the maximum and minimum value is 4\/15<\/p>\n

2)\u00a0Answer\u00a0(D)<\/strong><\/p>\n

g(x) reaches its maximum when |x+8| =0 or x = -8. So, maximum value of g(x) is 22<\/p>\n

3)\u00a0Answer\u00a0(C)<\/strong><\/p>\n

$\\frac{y+x}{y+2x}$ = 1- $\\frac{x}{y+2x}$, which equals 1 – $\\frac{1}{\\frac{y}{x}+2}$. So, the function is at its maximum when y\/x is at its highest (x=2, y= 5) and is at its minimum when y\/x is lowest (x=3, y= 4).<\/p>\n

4)\u00a0Answer\u00a0(A)<\/strong><\/p>\n

$\\frac{y+7x}{y+4x}$ = 1 + $\\frac{3x}{y+4x}$, which equals 1 – $\\frac{3}{y\/x+4}$. This reaches its maximum when $\\frac{y}{x}$ is at its minimum and vice versa. So, the minimum value of $\\frac{y+7x}{y+4x}$ is $\\frac{5+7}{5+4}$ = 4\/3 and the maximum is $\\frac{1+28}{1+16}$ = 29\/17<\/p>\n

5)\u00a0Answer\u00a0(B)<\/strong><\/p>\n

$X+X+X+Y+Y+Y+Y = 14.$ So,$(X+X+X+Y+Y+Y+Y)\/7 \\geq (X*X*X*Y*Y*Y*Y)^{1\/7}$ So, $(X^{3}*Y^{4})^{1\/7} \\leq 2$. So, the maximum value of $X^{3}*Y^4$ is 128<\/p>\n

6)\u00a0Answer\u00a0(B)<\/strong><\/p>\n

If $x \\leq -11$, g(x) = 45.
\nIf $-11 < x \\leq 34$, g(x) = 23-2x which peaks when x= -11.
\nIf $x> 34$, g(x) = -45.
\nSo, the maximum value of g(x) is 45<\/p>\n

7)\u00a0Answer\u00a0(B)<\/strong><\/p>\n

$X\/3 + X\/3 +X\/3 + Y\/2 + Y\/2 = 10.$ So, $X^{3}*Y^{2} \\leq 2^{5} * 3^{3}*2^{2} = 3456.$ The equality happens when X=6 and Y=4<\/p>\n

8)\u00a0Answer\u00a0(C)<\/strong><\/p>\n

f(x) reaches its maximum value when 3x-4 = 11-7x or if x = 3\/2 So, the maximum value of f(x) is 1\/2<\/p>\n

9)\u00a0Answer\u00a0(A)<\/strong><\/p>\n

F(x) attains its maximum value when X = 0. So, the number of real values for which F(X) attains its minimum value is 1<\/p>\n

10)\u00a0Answer\u00a0(D)<\/strong><\/p>\n

There are twelve integral solutions to the equation |a-2|+|b-3| = 4 The product a*b is maximized when a=4 and b=5 and equals 20<\/p>\n

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11)\u00a0Answer\u00a0(D)<\/strong><\/p>\n

The minimum value of W is when Y=4, X = -2 and Z=3 and equals -2\/3 The maximum value of W is when Y=-4, X=-2 and Z=3 and equals 2\/3<\/p>\n

12)\u00a0Answer\u00a0(C)<\/strong><\/p>\n

The maximum value occurs when 2X and Y are closest to each other. So, when X=8 and Y=17, XY reaches its peak of 136.<\/p>\n

13)\u00a0Answer\u00a0(C)<\/strong><\/p>\n

The maximum value occurs when 3A and 2B are closest to each other. So, when A=10 and B=17, AB reaches its peak of 170.<\/p>\n

14)\u00a0Answer\u00a0(A)<\/strong><\/p>\n

There are fifteen solutions to the equation |a-1|+|b-2|+|c-3|=3.
\nLet |a-1| be x, |b-2| be y and |c-3| be z.
\nx, y and z are whole numbers.
\nx + y + z = 3.
\nTherefore, the number of solutions to this equation will be n+r-1 C r-1 = 3+3-1C2 = 5C2 = 10.
\nThere are 10 solutions to the equation x+y+z = 3.
\nAfter enumerating them, we find that (a+1)(b+2)(c+3) is maximum when a=b=c=3 and its maximum value is 120.<\/p>\n

15)\u00a0Answer\u00a0(B)<\/strong><\/p>\n

a*(1\/b+1\/c)+b*(1\/c+1\/a)+c*(1\/a+1\/b) equals (a\/b+b\/a)+(b\/c+c\/b)+(a\/c+c\/a).
\nAll these terms are at their minimum when a = b = c and the minimum value of the sum is 6<\/p>\n

Option b) is the correct answer.<\/p>\n

16)\u00a0Answer\u00a0(A)<\/strong><\/p>\n

$\\frac{y+7x}{y+4x}$ = 1 + $\\frac{3x}{y+4x}$, which equals 1 + $\\frac{3}{y\/x+4}$.<\/p>\n

This reaches its maximum when $\\frac{y}{x}$ is at its minimum and vice versa.<\/p>\n

So, the minimum value of $\\frac{y+7x}{y+4x}$ is $\\frac{5+7}{5+4}$ = 4\/3 and the maximum is $\\frac{1+28}{1+16}$ = 29\/17<\/p>\n

17)\u00a0Answer\u00a0(B)<\/strong><\/p>\n

Taking lowest possible positive value of m i.e. 1 . Such that a+b+c+d=5 , so atleast one of them must be grater than 1 ,<\/p>\n

take a=b=c=1 and d=2<\/p>\n

we get $a^2 + b^2 + c^2 + d^2 = 7$ which is equal to $4m^2+2m+1$ for other values it is greater than $4m^2+2m+1$ . so option B<\/p>\n

18)\u00a0Answer\u00a0(C)<\/strong><\/p>\n

For the given expression value of x,y,z are distinct positive integers . So the value of expression will always be greater than value when all the 3 variables are equal . substitute x=y=z we get minimum value of 6 .<\/p>\n

$(x^2(y+z) + y^2(x+z) + z^2(x+y))\/xyz$ = x\/z + x\/y + y\/z + y\/x + z\/y + z\/x<\/p>\n

Applying AM greater than or equal to GM, we get minimum sum = 6<\/p>\n

 <\/p>\n

19)\u00a0Answer\u00a0(C)<\/strong><\/p>\n

Since the product is constant,<\/p>\n

(a+b+c+d)\/4 >= $(abcd)^{1\/4}$<\/p>\n

We know that abcd = 1.<\/p>\n

Therefore, a+b+c+d >= 4<\/p>\n

$(a+1)(b+1)(c+1)(d+1)$
\n= $1+a+b+c+d+ ab + ac + ad + bc + bd + cd+ abc+ bcd+ cda+ dab+abcd$
\nWe know that $abcd = 1$
\nTherefore, $a = 1\/bcd, b = 1\/acd, c = 1\/bda$ and $d = 1\/abc$
\nAlso, $cd = 1\/ab, bd = 1\/ac, bc = 1\/ad$
\nThe expression can be clubbed together as $1 + abcd + (a + 1\/a) + (b+1\/b) + (c+1\/c) + (d+1\/d) + (ab + 1\/ab) + (ac+ 1\/ac) + (ad + 1\/ad)$
\nFor any positive real number $x$, $x + 1\/x \\geq 2$
\nTherefore, the least value that $(a+1\/a), (b+1\/b)….(ad+1\/ad)$ can take is 2.
\n$(a+1)(b+1)(c+1)(d+1) \\geq 1 + 1 + 2 + 2 + 2+ 2 + 2 + 2 + 2$
\n=> $(a+1)(b+1)(c+1)(d+1) \\geq 16$
\nThe least value that the given expression can take is 16. Therefore, option C is the right answer.<\/p>\n

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20)\u00a0Answer\u00a0(C)<\/strong><\/p>\n

Approach 1:<\/p>\n

The given expression is symmetric in x and y and the limiting condition (x+y=1) is also symmetric in x and y.<\/p>\n

=>This means that the expression attains the minimum value when x = y
\nx=y=1\/2
\nSo, the value = $(x+\\frac{1}{x})^2+(y+\\frac{1}{y})^2$ = $(2+\\frac{1}{2})^2+(2+\\frac{1}{2})^2$ =12.5<\/p>\n

Approach 2:<\/p>\n

$(x+1\/x)^2$ + $(y+1\/y)^2$ = $(x+1\/x+y+1\/y)^2$ – $2*(x+1\/x)(y+1\/y)$<\/p>\n

Let x+1\/x and y+1\/y be two terms. Thus (x+1\/x+y+1\/y)\/2 would be their Arithmetic Mean(AM) and $\\sqrt{(x+1\/x)(y+1\/y)}$ would be their Geometric Mean (GM).<\/p>\n

Therefore, we can express the above equation as $(x+1\/x)^2$ + $(y+1\/y)^2$ = $4AM^2$ – $GM^2$. As AM >= GM, the minimum value of expression would be attained when AM = GM.<\/p>\n

When AM = GM, both terms are equal. That is x+1\/x = y +1\/y.<\/p>\n

Substituting y=1-x we get<\/p>\n

x+1\/x = (1-x) + 1\/(1-x)<\/p>\n

On solving we get 2x-1 = (2x-1)\/ x(1-x)<\/p>\n

So either 2x-1 = 0 or x(1-x) = 1<\/p>\n

x(1-x) = x * y<\/p>\n

As x and y are positive numbers whose sum = 1, 0<= x, y <=1. Hence, their product cannot be 1.<\/p>\n

Thus, 2x-1 = 0 or x=1\/2<\/p>\n

=> y = 1\/2<\/p>\n

So, the value = $(x+\\frac{1}{x})^2+(y+\\frac{1}{y})^2$ = $(2+\\frac{1}{2})^2+(2+\\frac{1}{2})^2$ = 12.5<\/p>\n

21)\u00a0Answer\u00a0(C)<\/strong><\/p>\n

Since the product is constant, the expression (1+a)(1+b)(1+c)(1+d) is least when a = b = c = d.
\nSo, a = b = c = d = 1 and the minimum value of the expression is 2*2*2*2 = 16.<\/p>\n

22)\u00a0Answer\u00a0(C)<\/strong><\/p>\n

The expression attains the minimum value when x = y
\nx=y=1\/2
\nSo, the value = $2.5^2 * 2 = 12.5$<\/p>\n

23)\u00a0Answer\u00a0(B)<\/strong><\/p>\n

AM >= HM<\/p>\n

So, (a+b+c+d)\/4 >= 4\/(1\/a+1\/b+1\/c+1\/d) => (a+b+c+d)*(1\/a+1\/b+1\/c+1\/d) >= 4*4 = 16<\/p>\n

24)\u00a0Answer\u00a0(D)<\/strong><\/p>\n

| r-6 | = 11 => r = -5 or 17<\/p>\n

| 2q – 12 | = 8 => q = 10 or 2<\/p>\n

So, the minimum possible value of q\/r = 10\/(-5) = -2<\/p>\n

25)\u00a0Answer\u00a0(C)<\/strong><\/p>\n

The mean of $\\frac{1}{3}$ power of two numbers is less than $\\frac{1}{3}$ power of the mean of the numbers.
\nHence, $\\frac{p^{\\frac{1}{3}}+q^{\\frac{1}{3}}}{2} \\leq {(\\frac{p+q}{2})}^{\\frac{1}{3}}$
\nSo, $p^\\frac{1}{3}+q^\\frac{1}{3} \\leq 2^\\frac{2}{3}$<\/p>\n

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Maxima and Minima Questions for CAT: Download important CAT Maxima and Minima Questions PDF based on previous asked questions in CAT and other MBA exams. Top 25 Maxima and Minima functions questions for CAT quantitative aptitude. 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