In interval [1,e], the greatest value of ${x^2}\log x$ is

$\frac{1}{e}\log \frac{1}{{\sqrt e }}$

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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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31 Hard

31 of 174

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.

32 Easy

32 of 174

Application +4 | -1

For which value of

x

f(x)=\sum\limits_{{k=1}}(x-k)^2

minimum ?

View answer and explanation

Correct option: B

1008

Option B (1008) is correct because by taking the derivative of the function and setting it to zero, we find the value of x that minimizes the function.

33 Easy

33 of 174

Understanding +4 | -1

A function

f

such that

f^{\prime}(2)=f^{\prime \prime}(2)=0

and

f

has a local maximum of -17 at 2 is

View answer and explanation

Correct option: C

-17-(x-2)^{4}

Option C (

-17-(x-2)^{4}

) is correct because it satisfies the given conditions:

f'(2) = f''(2) = 0

and has a local maximum of -17 at x=2.

34 Easy

34 of 174

Application +4 | -1

The minimum value of

f(x)=|3-x|+|2+x|+|5-x|

View answer and explanation

Correct option: B

Option B (7) is correct because the minimum value of the function occurs at x=3, and f(3) = 7.

35 Medium

35 of 174

Application +4 | -1

The curve

y(x)=a x^3+b x^2+c x+5

touches the

x

-axis at the point

P(-2,0)

and cuts the

y

-axis at the point

Q

, where

y^{\prime}

is equal to 3 . Then the local maximum value of

y(x)

View answer and explanation

Correct option: A

\frac{27}{4}

Option C (

\frac{{33}}{{2^{28}}}

) is correct because it represents the probability of one or two successes, calculated using the binomial distribution formula and the given mean and variance.

36 Medium

36 of 174

Analyzing +4 | -1

If the greatest and least values of the function

f(x)

=x^3-6 x^2+9 x+1

[0,2]

are

\lambda

and

\mu

, then the value of

\lambda^4+\mu^8

must be:

View answer and explanation

Correct option: A

626

Option A (626) is correct because the greatest and least values of the function on the given interval are 5 and 1, respectively, and 5⁴ + 1⁸ = 626.

37 Easy

37 of 174

Analyzing +4 | -1

If the curve

x^2=-4(y-a)

does not intersect the curve

y=\left[x^2-x+1\right] (

where [.] denotes the greatest integer function) in

\left[0, \frac{1+\sqrt{5}}{2}\right]

, then

View answer and explanation

Correct option: C

\frac{1}{4} < a < 1

Option C (

\frac{1}{4} < a < 1

) is correct. By analyzing the intersection conditions of the parabola and the greatest integer function, we find the range of

a

to be

\frac{1}{4} < a < 1

38 Medium

38 of 174

Analyzing +4 | -1

The least distance of the point

Q(0,-2)

from the point

P(x, y)

where

y=\frac{16}{\sqrt{3} x^3}-2

and

x>0

View answer and explanation

Correct option: C

4 / \sqrt{3}

Option C (

4 / \sqrt{3}

) is correct because it is the minimum distance calculated by finding the critical point of the distance function and verifying it's a minimum.

39 Medium

39 of 174

Analyzing +4 | -1

The distance of that point on

y=x^4+3 x^2+2 x

which is nearest to the line

y=2 x-1

View answer and explanation

Correct option: D

\frac{1}{\sqrt{5}}

Option D (

\frac{{1}}{{\sqrt{{5}}}}

) is correct because it represents the minimum distance found by setting the derivative of the distance function to zero.

40 Medium

40 of 174

Application +4 | -1

a{x^2} + bx + 4

attains its minimum value

- 1

x = 1,

then the values of

a\,and\,b

are respectively

View answer and explanation

Correct option: A

5, - 10

Option A (5, -10) is correct because these values satisfy the conditions for the quadratic function to attain a minimum value of -1 at x=1.

41 Medium

41 of 174

Analyzing +4 | -1

The minimum value of

2{\log _{10}}x - {\log _x}0.01,\;x > 1

, is

View answer and explanation

Correct option: D

\frac{1}{2}

Option D is correct because the minimum value is found by applying AM-GM inequality after simplifying the expression.

42 Easy

42 of 174

Application +4 | -1

The function

f(x)=\frac{x}{2}+\frac{2}{x}

has a local minimum at

View answer and explanation

Correct option: A

x=2

Option A (

x=2

) is correct because the second derivative of the function is positive at this point, indicating a local minimum.

43 Easy

43 of 174

Understanding +4 | -1

If for a function

f\left( x \right),\;f'\left( a \right) = 0,\;f''\left( a \right) = 0,\;f'''\left( a \right) > 0

, then at

x = a,\;f\left( x \right)

View answer and explanation

Correct option: C

Not an extreme point

Option C (Not an extreme point) is correct because if the first and second derivatives are zero at a point, and the third derivative is non-zero, then the point is a point of inflection, not an extreme point.

44 Medium

44 of 174

Application +4 | -1

x - 2y = 4

, the minimum value of

xy

View answer and explanation

Correct option: A

- 2

Option A (

- 2

) is correct because it is the minimum value of

xy

given the constraint

x - 2y = 4

, found by substitution and differentiation.

45 Medium

45 of 174

Analyzing +4 | -1

Let

f(x)=\frac{x^2-1}{x^2+1}, x \in \mathbb{R}

. Then the minimum value of

f

View answer and explanation

Correct option: D

is equal to -1

Option D (is equal to -1) is correct because the minimum value of the function is -1, which occurs at x=0.

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46 Easy

46 of 174

Analyzing +4 | -1

\sin x+\cos x=y^2-y+a

has no value of

x

for any

y

, if

a

belongs

View answer and explanation

Correct option: D

(\sqrt{3}, \infty)

Option D (

(\sqrt{3}, \infty)

) is correct because the range of

\sin x + \cos x

[-\sqrt{2}, \sqrt{2}]

, and

y^2 - y + a

must be outside this range for all

y

47 Easy

47 of 174

Analyzing +4 | -1

In interval [1,e], the greatest value of

{x^2}\log x

View answer and explanation

Correct option: A

{e^2}

The correct answer is A (

{e^2}

) because it is the greatest value of the function

{x^2}\log x

in the interval [1,e].

48 Hard

48 of 174

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.

49 Easy

49 of 174

Analyzing +4 | -1

Let

f:(-\infty, \infty) \rightarrow(-\infty, \infty)

be defined by

f(x)=x^3+1

Statement 1: The function

f

has a local extremum at

x=0

Statement 2: The function

f

is continuous and differentiable on

(-\infty, \infty)

and

f^{\prime}(0)=0

View answer and explanation

Correct option: D

Statement 1 is false, Statement 2 is true.

Statement 1 is false, and Statement 2 is true. The function f(x) = x³ + 1 does not have a local extremum at x=0, but it is continuous and differentiable with f'(0) = 0.

50 Hard

50 of 174

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.

51 Easy

51 of 174

Analyzing +4 | -1

The maximum value of

{\left( {1/x} \right)^x}

View answer and explanation

Correct option: C

{e^{1/e}}

Option C (

{e^{1/e}}

) is correct because it represents the maximum value of the function found by analyzing its derivatives.

52 Medium

52 of 174

Application +4 | -1

N

characters of information are held on magnetic tape, in batches of

x

characters each, the batch processing time is

\alpha + \beta {x^2}

seconds,

\alpha \,and\,\beta

are constants. The optical value of

x

for fast processing is,

View answer and explanation

Correct option: C

\sqrt {\alpha /\beta }

Option C (

\sqrt {\alpha /\beta }

) is correct because it represents the value of x that minimizes the total processing time, found by taking the derivative and setting it to zero.

53 Medium

53 of 174

Analyzing +4 | -1

If the function

f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1

attains its maximum and minimum at

p

and

q

respectively such that

{p^2} = q

, then a equals

View answer and explanation

Correct option: C

Option C (2) is correct because it satisfies the given conditions for maximum and minimum points and the relation between them.

54 Medium

54 of 174

Application +4 | -1

The minimum value of

4{e^{2x}} + 9{e^{ - 2x}}

View answer and explanation

Correct option: B

Option B (12) is correct because it represents the minimum value of the given expression, found by using calculus to minimize the function or by applying AM-GM inequality.

55 Hard

55 of 174

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.

56 Easy

56 of 174

Analyzing +4 | -1

The maximum value of

\frac{{\log x}}{x}\,\mathrm{in}\,\left( {2,\infty } \right),\,\mathrm{is}

View answer and explanation

Correct option: D

\frac{1}{e}

Option D (

\frac{1}{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.

57 Medium

57 of 174

Application +4 | -1

For a certain curve

\frac{d^{2} y}{d x^{2}}=6 x-4

and curve has local minimum value 5 at

x=1

. Let the global maximum and global minimum values, where

0 \leq x \leq 2

; are

M

and

m

. Then the value of

(M-m)

equals to :

View answer and explanation

Correct option: B

Option B (2) is correct because finding the first derivative, setting it to zero, and using the given minimum value allows us to determine the function and its maximum and minimum values in the given range.

58 Medium

58 of 174

Application +4 | -1

The function

f\left( x \right) = a\sin x + \frac{1}{3}\sin 3x

has maximum value at

x = \frac{\pi }{3}

. The value of

a

View answer and explanation

Correct option: C

Option C (2) is correct because it is the value of 'a' that makes the derivative of the function equal to zero at x = π/3, indicating a maximum value.

59 Medium

59 of 174

Analyzing +4 | -1

If the maximum value of

a

, for which the function

f_a(x)=\tan ^{-1} 2 x-3 a x+7

is non-decreasing in

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

, is

\bar{a}

, then

f_{\bar{a}}\left(\frac{\pi}{8}\right)

is equal to

View answer and explanation

Correct option: A

8-\frac{9 \pi}{4\left(9+\pi^2\right)}

Option A (

8-\frac{9 \pi}{4(9+\pi^2)}

) is correct because it involves finding the maximum value of 'a' for which the function is non-decreasing and then evaluating the function at

\frac{\pi}{8}

60 Medium

60 of 174

Analyzing +4 | -1

\max \limits_{{0 \leq x \leq \pi}}\left\{x-2 \sin x \cos x+\frac{1}{3} \sin 3 x\right\}=

View answer and explanation

Correct option: D

\frac{5 \pi+2+3 \sqrt{3}}{6}

Option D (

\frac{5 \pi+2+3 \sqrt{3}}{6}

) is correct because it represents the maximum value of the function within the given interval, found by analyzing the critical points.

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