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 · Medium

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Showing 1-30 of 97 questions

1 Medium

1 of 97

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 Medium

2 of 97

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

3 Medium

3 of 97

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}

4 Medium

4 of 97

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.

5 Medium

5 of 97

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.

6 Medium

6 of 97

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.

7 Medium

7 of 97

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.

8 Medium

8 of 97

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.

9 Medium

9 of 97

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.

10 Medium

10 of 97

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.

11 Medium

11 of 97

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.

12 Medium

12 of 97

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.

13 Medium

13 of 97

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.

14 Medium

14 of 97

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.

15 Medium

15 of 97

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.

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16 Medium

16 of 97

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.

17 Medium

17 of 97

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}}

18 Medium

18 of 97

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.

19 Medium

19 of 97

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.

20 Medium

20 of 97

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.

21 Medium

21 of 97

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.

22 Medium

22 of 97

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.

23 Medium

23 of 97

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.

24 Medium

24 of 97

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.

25 Medium

25 of 97

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.

26 Medium

26 of 97

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.

27 Medium

27 of 97

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.

28 Medium

28 of 97

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.

29 Medium

29 of 97

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.

30 Medium

30 of 97

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.

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Key Takeaways

✓ This set contains 97 single-correct-answer MCQs on MAXIMA & MINIMA.
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