Let the function $ f(x)=2 x^{3}+(2 p-7) x^{2}+3(2 p-9) x $ -6 have a maxima for some value of $ x 0 $. Then, the set of all values of $ p $ is

$ \left(0, \frac{9}{2}\right) $

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

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

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JEE • MATHEMATICS • MAXIMA & MINIMA

Maxima & minima practice · Mixed difficulty

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 1-30 of 174 questions

1 Medium

1 of 174

Analyzing +4 | -1

Let

f(x)=\left(1+b^2\right) x^2+2 b x+1

. Let

m(b)

be the minimum value of

f(x)

(that is least value). As

b

varies, the range of

m(b)

View answer and explanation

Correct option: D

(0,1]

Option D ((0,1]) is correct because the range of the minimum value function m(b) is calculated using calculus and analysis of the function's behavior.

2 Easy

2 of 174

Analyzing +4 | -1

Let

f: R \rightarrow R

be a function defined by

f(x)=(x-3)^{n_1}(x-5)^{n_2}, n_1, n_2 \in N

. Then, which of the following is NOT true?

View answer and explanation

Correct option: C

For

n_1=3, n_2=5

, there exists

\alpha \in(3,5)

where

f

attains local maxima.

The correct answer is C. For

n_1=3, n_2=5

, there exists

\alpha \in(3,5)

where

f

attains local maxima is NOT true because with both powers odd, there will be a local minimum.

3 Medium

3 of 174

Analyzing +4 | -1

The equation

|2 a x-3|+|a x+1|+|5-a x|=\frac{1}{2}

possesses

View answer and explanation

Correct option: D

no real solutions for all

a \in R

Option D (no real solutions for all

a \in R

) is correct. The inequality derived from the triangle inequality shows that the equation has no real solutions for any value of

a

4 Medium

4 of 174

Analyzing +4 | -1

Let the function

f(x)=2 x^{3}+(2 p-7) x^{2}+3(2 p-9) x

-6 have a maxima for some value of

x<0

and a minima for some value of

x>0

. Then, the set of all values of

p

View answer and explanation

Correct option: D

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

Option D (

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

) is correct because the condition for maxima at x < 0 and minima at x > 0 implies that

f'(0) < 0

, leading to

p < \frac{9}{2}

5 Medium

5 of 174

Analyzing +4 | -1

The least value of the

f\left( x \right)

given by

f\left( x \right) = {\tan ^{ - 1}}x - \frac{1}{2}{\log _e}x

in the interval

\left[ {1/\sqrt 3 ,\;\sqrt 3 } \right]

, is

View answer and explanation

Correct option: B

\frac{\pi }{3} - \frac{1}{4}{\log _e}3

Option B (

\frac{\pi }{3} - \frac{1}{4}{\log _e}3

) is correct because it represents the least value of the function within the specified interval, determined by evaluating the function at critical points and endpoints.

6 Hard

6 of 174

Analyzing +4 | -1

If the total maximum value of the function

f(x)=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^{2} x}, x \in\left(0, \frac{\pi}{2}\right),

\frac{k}{e}

, then

\left(\frac{k}{e}\right)^{8}+\frac{k^{8}}{e^{5}}+k^{8}

is equal to

View answer and explanation

Correct option: A

e^{3}+e^{6}+e^{11}

Option A (

e^{3}+e^{6}+e^{11}

) is correct because the maximum value of the function is found to be

e^{3/8}

, leading to the given expression.

7 Easy

7 of 174

Application +4 | -1

The sum of two numbers is 20. If the product of the square of one number and cube of the other is maximum, then the numbers are

View answer and explanation

Correct option: A

12, 8

Option A (12, 8) is correct because these numbers maximize the product of the square of one number and the cube of the other, given their sum is 20.

8 Medium

8 of 174

Analyzing +4 | -1

Let

f(x)=\left|x^2-5 x+6\right|, x \in[0,2.4]

. Then, the sum of the maximum and minimum values of

f

[0

, 2.4] is

View answer and explanation

Correct option: A

Option A (6) is correct because the sum of the maximum and minimum values of f on [0, 2.4] is 6.

9 Easy

9 of 174

Analyzing +4 | -1

The coordinates of the point on the curve

x^3=y(x-a)^2, a>0

, where the ordinate is minimum

View answer and explanation

Correct option: C

\left(3 a, \frac{27 a}{4}\right)

Option C

\left(3 a, \frac{{27 a}}{{4}}\right)

is correct because it represents the point where the first derivative is zero and the second derivative is positive, indicating a minimum ordinate.

10 Medium

10 of 174

Analyzing +4 | -1

Absolute value of difference of maximum and minimum values of

{x^3} - 18{x^2} + 96x

in interval

(0, 9)

is _____

View answer and explanation

Correct option: A

Option A is correct because the absolute difference between the maximum and minimum values of the function in the given interval is 32.

11 Medium

11 of 174

Analyzing +4 | -1

The maximum value of

P=\frac{2}{a^2+1}-\frac{2}{b^2+1}+\frac{3}{c^2+1}

is ______ where

{a}, {b}, {c}>0

and

a b c+a+c=b

View answer and explanation

Correct option: D

\frac{10}{3}

Option D is correct because the maximum value is found by using trigonometric substitutions and maximizing the resulting expression using calculus or completing the square.

12 Medium

12 of 174

Analyzing +4 | -1

The greatest value of the function

f(x)=\frac{x}{a x^2+b}

View answer and explanation

Correct option: B

\frac{1}{2 \sqrt{a b}}

Option B (

\frac{1}{2 \sqrt{a b}}

) is correct because it represents the maximum value of the function, found by using calculus to determine the critical points.

13 Medium

13 of 174

Application +4 | -1

Let

x, y, z

be positive numbers such that

x+y+z=26 \text { and } y=3 x

x^2+y^2+z^2

is to be least, then the value of

x

View answer and explanation

Correct option: B

The correct answer is B (4) because the minimum value of

x^2 + y^2 + z^2

occurs when

x = 4

, which is found by minimizing the function using calculus.

14 Medium

14 of 174

Analyzing +4 | -1

If a differentiable function

f\left( x \right)

has a relative minimum at

x = 0

, then the function

\phi \left( x \right) = f\left( x \right) + ax + b

has a relative minimum at

x = 0

for

View answer and explanation

Correct option: B

All b if

a = 0

Option B (All b if a = 0) is correct because if a=0, the derivative of φ(x) at x=0 is determined by f'(x), which has a relative minimum at x=0.

15 Medium

15 of 174

Analyzing +4 | -1

Let

f: \mathrm{R} \rightarrow \mathrm{R}

be a continuous function defined by

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

Statement -1:

f(c)=\frac{1}{3}

, for some

c \in \mathrm{R}

.
Statement -2:0 < f(x)

\leq \frac{1}{2 \sqrt{2}}

, for all

x \in \mathrm{R}

View answer and explanation

Correct option: D

Statement - 1 is true, Statement 2 is true ; Statement -2 is a correct explanation for Statement -1 .

Statement - 1 is true, Statement - 2 is true, and Statement -2 is a correct explanation for Statement -1. The maximum value of f(x) is 1/(2*sqrt(2)), and since 1/3 is less than this maximum, f(c) = 1/3 for some c.

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16 Easy

16 of 174

Analyzing +4 | -1

On the interval

\left[\frac{5 \pi}{4}, \frac{4 \pi}{3}\right]

, the least value of the function

f(x)=\int_{5 \pi / 4}^x(3 \sin t+4 \cos t) d t

View answer and explanation

Correct option: A

\frac{3}{2}+\frac{1}{\sqrt{2}}-2 \sqrt{3}

Option A is correct because the function

f(x)

is decreasing on the given interval, so the least value occurs at the right endpoint,

x = \frac{4\pi}{3}

17 Easy

17 of 174

Analyzing +4 | -1

\left|a_2\right| > \left|a_1+a_3\right|

, then atleast one root of the equation

a_1 x^4+a_2 x^3+a_3=0

lies in the interval

View answer and explanation

Correct option: B

(-1,1)

Option B ((-1,1)) is correct because the condition |a_2| > |a_1+a_3| implies that there is a sign change in the function within the interval (-1, 1), indicating a root.

18 Medium

18 of 174

Application +4 | -1

f:[0,4] \rightarrow \mathrm{R}

is a differentiable function. If there exist

a, b \in(0,4)

such that

(f(4))^2-(f(0))^2=k f^{\prime}(a) f(b)

, then

k=\ldots \ldots

View answer and explanation

Correct option: A

Option A (8) is correct because by applying the Mean Value Theorem and manipulating the resulting equations, the value of k is found to be 8.

19 Medium

19 of 174

Application +4 | -1

The sum of an infinitely decreasing geometric progression is equal to the least value of the function

f(x)=3 x^2-x+\frac{25}{12}

and the first term of the progression is equal to the square of its common ratio. Then, the common ratio is

View answer and explanation

Correct option: B

\sqrt{3}-1

Option B (

\sqrt{{3}}-1

) is correct because it is the common ratio of the geometric progression, derived by minimizing the function and solving the resulting quadratic equation.

20 Medium

20 of 174

Application +4 | -1

Let

2 p+3 q+4 r=15

and if the expression

p^3 q^5 r^7

takes the maximum value, then

p: q: r

is equal to (where

{p}, {q}, {r} \in {R}^{+}

View answer and explanation

Correct option: A

18: 20: 21

Option A (18:20:21) is correct because to maximize the expression p³q⁵r⁷ subject to the constraint 2p+3q+4r=15, we need to set 2p/3 = 3q/5 = 4r/7, leading to the ratio 18:20:21.

21 Medium

21 of 174

Application +4 | -1

\alpha \in\left(0, \frac{\pi}{2}\right)

; then minimum value of

\sqrt{x^2+x}+\frac{\tan ^2 \alpha}{\sqrt{x^2+x}}

View answer and explanation

Correct option: C

2 \tan \alpha

Option C (

2 \tan \alpha

) is correct because it represents the minimum value obtained by applying the AM-GM inequality to the given expression.

22 Medium

22 of 174

Analyzing +4 | -1

The function

f(x)=x+\tan x

has

View answer and explanation

Correct option: D

neither maximum nor minimum

Option D (neither maximum nor minimum) is correct because the derivative of the function is always positive, indicating that the function is strictly increasing.

23 Easy

23 of 174

Analyzing +4 | -1

Let

f\left( x \right) = \left\{ {\begin{array}{cccccccccccccccccccc} {\left| {{x^3} + {x^2} + 3x + \sin x} \right|\left( {3 + \sin \frac{1}{x}} \right),\;x \ne 0}\\ {0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x = 0} \end{array}} \right.

then number of points (where

f\left( x \right)

attains its minimum value) is

View answer and explanation

Correct option: A

Option A (1) is correct because the function attains its minimum value of 0 only at x=0, due to the absolute value and the term (3 + sin(1/x)).

24 Medium

24 of 174

Analyzing +4 | -1

The maximum value of

{x^{1/x}}

View answer and explanation

Correct option: C

{e^{1/e}}

Option C (

{e^{1/e}}

) is correct because it is the maximum value of the function

{x^{1/x}}

25 Hard

25 of 174

Analyzing +4 | -1

f'\left( x \right) = {\left( {x - a} \right)^{2n}}{\left( {x - b} \right)^{2\;p + 1}}

where

n

and

p

are positive integers, then

View answer and explanation

Correct option: C

x = a

is not a point of maximum or minimum

Option C (

x = a

is not a point of maximum or minimum) is correct because the sign of the derivative does not change as

x

passes through

a

26 Medium

26 of 174

Analyzing +4 | -1

The number of values of

x

at which the function

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

has extremum values is

View answer and explanation

Correct option: C

Option C (2) is correct because the function has two critical points, x=0 and x=2/5, where it attains extremum values.

27 Medium

27 of 174

Analyzing +4 | -1

Let

f(x)=3 \sin x-4 \sin ^3 x

. The length of the longest interval in which

f^{\prime}(x) \geq 0

View answer and explanation

Correct option: A

\frac{\pi}{3}

Option A (

\frac{π}{3}

) is correct because it represents the length of the interval where the derivative of f(x) is non-negative.

28 Hard

28 of 174

Analyzing +4 | -1

A function

f

is defined by

f(x)=|x|^m|x-1|^n, \forall x \in R

. The maximum value of the function is

(m, n \in N)

View answer and explanation

Correct option: C

\frac{m^m \cdot n^n}{(m+n)^{m+n}}

Option C (

\frac{m^m \cdot n^n}{(m+n)^{m+n}}

) is correct. By finding the critical points of the function and evaluating the function at those points, we find the maximum value to be

\frac{m^m \cdot n^n}{(m+n)^{m+n}}

29 Medium

29 of 174

Analyzing +4 | -1

The maximum value of

x^4 e^{-x^2}

is:

View answer and explanation

Correct option: D

4 e^{-2}

Option D (

4 e^{-2}

) is correct because it represents the maximum value of the function obtained by finding the critical points and using the second derivative test.

30 Easy

30 of 174

Analyzing +4 | -1

If the absolute maximum value of the function

f(x)=

\left(x^2-2 x+7\right) e^{\left(4 x^3-12 x^2-180 x+31\right)}

in the interval

[-3,0]

f(\alpha)

, then

View answer and explanation

Correct option: B

\alpha=-3

Option B (

\alpha=-3

) is correct because the absolute maximum value of the function occurs at x = -3 within the given interval [-3, 0].

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Key Takeaways

✓ This set contains 174 single-correct-answer MCQs on MAXIMA & MINIMA.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

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