Let f x = 2 x + tan - 1 x and g x = log e 1 + x 2 + x , x ∈ 0 , 3 . Then

There exists x ∈ 0 , 3 such that f ' x < g ' x

There exist 0 < x 1 < x 2 < 3 such that f x < g x , ∀ x ∈ x 1 , x 2

HARD

JEE Main

IMPORTANT

Earn 100

A wire of length $36 m$ is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is $k$ (meter), then $(\frac{4}{π} + 1) k$ is equal to

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Important Questions on Application of Derivatives

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 26 August 2021 Shift 2>Q 2

The local maximum value of the function,

f (x) = {(\frac{2}{x})}^{x^{2}}, x > 0,

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 26 August 2021 Shift 2>Q 3

Let

a

and

b

respectively be the points of local maximum and local minimum of the function

f (x) = 2 x^{3} - 3 x^{2} - 12 x

. If

A

is the total area of the region bounded by

y = f (x),

the

x

-axis and the lines

x = a

and

x = b,

then

4 A

is equal to ______.

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 27 August 2021 Shift 1>Q 4

A wire of length $20 m$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 27 August 2021 Shift 2>Q 5

A box open from top is made from a rectangular sheet of dimension $a \times b$ by cutting squares each of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum, then $x$ is equal to:

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 27 August 2021 Shift 2>Q 6

Let

M

and

m

respectively be the maximum and minimum values of the function

f (x) = \tan^{- 1} (\sin x + \cos x)

[0, \frac{π}{2}]

. Then the value of

\tan (M - m)

is equal to:

EASY

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 31 August 2021 Shift 1>Q 7

R

is the least value of

a

such that the function

f (x) = x^{2} + a x + 1

is increasing on

[1, 2]

and

S

is the greatest value of

a

such that the function

f (x) = x^{2} + a x + 1

is decreasing on

[1, 2]

, then the value of

| R - S |

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 31 August 2021 Shift 2>Q 8

An angle of intersection of the curves,

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

and

x^{2} + y^{2} = a b, a > b,

is :

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 31 August 2021 Shift 2>Q 9

Let

f (x)

be a cubic polynomial with

f (1) = - 10, f (- 1) = 6,

and has a local minima at

x = 1,

and

f^{'} (x)

has a local minima at

x = - 1 .

Then

f (3)

is equal to .

A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is k (meter), then 4π+1k is equal to

Important Questions on Application of Derivatives

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A wire of length $36 m$ is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is $k$ (meter), then $(\frac{4}{π} + 1) k$ is equal to