If $f(x)=\int_\limits {x^2}^{x^2} \frac{d t}{\log t}, x>0$, then

$f(x)$ is an increasing function $\forall x \in R^{+}$only

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JEE • MATHEMATICS • MAXIMA & MINIMA

Maxima & minima practice · Hard

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Practice format and chapter quiz

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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1 Hard

1 of 19

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.

2 Hard

2 of 19

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

3 Hard

3 of 19

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}}

4 Hard

4 of 19

Analyzing +4 | -1

a > b > 0

,then maximum value of

\frac{{ab\left( {{a^2} - {b^2}} \right)\sin x\cos x}}{{{a^2}{{\sin }^2}x + {b^2}{{\cos }^2}x}},\;x \in \left( {0,\;\pi /2} \right)

View answer and explanation

Correct option: B

\frac{{{a^2} - {b^2}}}{2}

Option B (

\frac{{{a^2} - {b^2}}}{2}

) is correct because it represents the maximum value of the given expression, achieved when the denominator is minimized.

5 Hard

5 of 19

Analyzing +4 | -1

A population

p\left( t \right)

of 1000 bacteria introduced into nutrient medium grows according to the relation

p\left( t \right) = 1000 + \frac{{1000t}}{{100 + {t^2}}}

. The maximum size of this bacterial population is

View answer and explanation

Correct option: C

1050

Option C (1050) is correct because it represents the maximum size of the bacterial population, found by analyzing the derivative of the population function.

6 Hard

6 of 19

Analyzing +4 | -1

The minimum & maximum value of

f(x)=\sin (\cos x)+\cos (\sin x)

for

-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}

are respectively

View answer and explanation

Correct option: A

\cos 1

and

1+\sin 1

Option A,

\cos 1

and

1+\sin 1

, is correct because the maximum of the function occurs at x=0 and the minimum occurs at x=

\pi/2

due to the decreasing nature of the function in the given interval.

7 Hard

7 of 19

Analyzing +4 | -1

The function

f\left( x \right) = ax + \frac{b}{x},\;b,\;x > 0

takes the least value at

x

equal to

View answer and explanation

Correct option: D

\sqrt {\frac{b}{a}}

Option D is correct because the function has a minimum at x =

\sqrt{{b/a}}

, which is found by setting the derivative equal to zero and confirming a positive second derivative.

8 Hard

8 of 19

Analyzing +4 | -1

The maximum value of

f\left( x \right) = \frac{x}{{4 + x + {x^2}}}

on [-1,1] is

View answer and explanation

Correct option: C

\frac{1}{6}

Option C (

\frac{1}{6}

) is correct because it represents the maximum value of the function on the given interval, found by evaluating the function at critical points and endpoints.

9 Hard

9 of 19

Analyzing +4 | -1

The maximum value of

{\left( {\frac{1}{x}} \right)^{2{x^2}}}

, is

View answer and explanation

Correct option: B

\sqrt[e]{e}

Option B (

\sqrt[e]{e}

) is correct because it represents the maximum value of the function, found by setting the derivative to zero and confirming it's a maximum.

10 Hard

10 of 19

Analyzing +4 | -1

Let

f\left( x \right) = 1 + 2{x^2} + {2^2}{x^4}\; + ... + {2^{10}}{x^{20}}

. Then,

f\left( x \right)

has

View answer and explanation

Correct option: A

More than one minimum

Option A (More than one minimum) is correct because the derivative of the function has multiple roots, indicating multiple minima.

11 Hard

11 of 19

Analyzing +4 | -1

The greatest value of

f\left( x \right) = {\left( {x + 1} \right)^{1/3}} - {\left( {x - 1} \right)^{1/3}}

on [0,1]is

View answer and explanation

Correct option: C

Option C (2) is correct. By finding the critical point and evaluating the function at the endpoints and critical point, we find the greatest value is 2.

12 Hard

12 of 19

Understanding +4 | -1

f(x)=\int\limits_{{x^2}}^{x^2} \frac{d t}{\log t}, x>0

, then

View answer and explanation

Correct option: D

f(x)

is neither maximum nor minimum at

x=1

Option D (

f(x)

is neither maximum nor minimum at

x=1

) is correct because the first derivative does not change sign at x=1.

13 Hard

13 of 19

Analyzing +4 | -1

The set of all values of the parameter

a

for which the points of minimum of the function

y = 1 + {a^2}x - {x^3}

satisfy the inequality

\frac{{{x^2} + x + 2}}{{{x^2} + 5x + 6}} \le 0

, is

View answer and explanation

Correct option: C

\left( {2\sqrt 3 ,\;3\sqrt 3 } \right)

The correct answer is C because the minimum point of the function must satisfy the given inequality, leading to the interval (2√3, 3√3).

14 Hard

14 of 19

Application +4 | -1

The minimum value of

px + qy

when

xy = {r^2}

, is

View answer and explanation

Correct option: A

2r\sqrt {pq}

Option A (

2r\sqrt {pq}

) is correct because it represents the minimum value obtained by minimizing the expression

px + qy

subject to the constraint

xy = r^2

using calculus.

15 Hard

15 of 19

Application +4 | -1

A cubic

f\left( x \right)

vanishes at

x = - 2

and has relative minimum/maximum at

x = - 1\,and\,x = \frac{1}{3}

such that

\int \limits_{ - 1}^1 f\left( x \right)dx = \frac{{14}}{3}

Then,

f\left( x \right)

View answer and explanation

Correct option: C

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

Option D (

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

) is correct because it satisfies the given conditions: vanishes at x = -2, has relative extrema at x = -1 and x = 1/3, and the integral from -1 to 1 is 14/3.

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16 Hard

16 of 19

Analyzing +4 | -1

A wire of length

20 \mathrm{m}

to be cut into two pieces. A piece of length

l_{1}

is bent to make a square of area

A_{1}

and the other piece of length

l_{2}

is made into a circle of area

A_{2}

. If

2 A_{1}+3 A_{2}

is minimum then

\left(\pi l_{1}\right): l_{2}

is equal to

View answer and explanation

Correct option: C

6: 1

Option C is correct because the ratio of π l_1 to l_2 is calculated to be 6:1 when the expression 2A_1 + 3A_2 is minimized.

17 Hard

17 of 19

Analyzing +4 | -1

The maximum value of

f\left( x \right) = \frac{x}{{4 + x + {x^2}}}

on [-1, 1] is

View answer and explanation

Correct option: D

\frac{1}{6}

Option D (

\frac{1}{6}

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

f(x)

on the given interval [-1, 1].

18 Hard

18 of 19

Analyzing +4 | -1

ax + \frac{b}{x} \ge c

for all positive values of

x

and a,b,c are positive constants, then

View answer and explanation

Correct option: A

ab \ge \frac{{{c^2}}}{4}

Option A (

ab \ge \frac{{{c^2}}}{4}

) is correct because the minimum value of the function

ax + \frac{b}{x}

must be greater than or equal to c, leading to the inequality

ab \ge \frac{{{c^2}}}{4}

19 Hard

19 of 19

Analyzing +4 | -1

The function

f\left( x \right) = 2{x^3} - 15{x^2} + 36x + 4

is maximum at

View answer and explanation

Correct option: A

x = 2

Option A (

x = 2

) is correct because the function's first derivative is zero and the second derivative is negative at this point, indicating a maximum.

Revision notes and key takeaways

Review MAXIMA & MINIMA in your textbook or class notes, then use this set to check your understanding.

Key Takeaways

✓ This set contains 19 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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