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

The sum of absolute maximum and absolute minimum values of the function $f (x) = |2 x^{2} + 3 x - 2| + \sin x \cos x$ in the interval $[0, 1]$ is

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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 - 24 June 2022 Shift 2>Q 106

Let the maximum area of the triangle that can be inscribed in the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{4} = 1, a > 2

, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the

y

-axis, be

6 \sqrt{3}

. Then the eccentricity of the ellipse is:

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 24 June 2022 Shift 2>Q 107

The number of distinct real roots of the equation

x^{7} - 7 x - 2 = 0

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 24 June 2022 Shift 2>Q 108

Let

λ^{*}

be the largest value of

λ

for which the function

f_{λ} (x) = 4 λ x^{3} - 36 λ x^{2} + 36 x + 48

is increasing for all

x \in ℝ

. Then

f_{λ^{*}} (1) + f_{λ, *} (- 1)

is equal to:

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 25 June 2022 Shift 1>Q 109

Let

f : R \to R

and

g : R \to R

be two functions defined by

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

and

g (x) = \frac{1 - 2 e^{2 x}}{e^{x}} \cdot

Then, for which of the following range of

α

, the inequality

f (g (\frac{{(α - 1)}^{2}}{3})) > f (g (α - \frac{5}{3}))

holds?

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 25 June 2022 Shift 2>Q 110

Water is being filled at the rate of

1 {cm}^{3} \sec^{- 1}

in a right circular conical vessel (vertex downwards) of height

35 cm

and diameter

14 cm

. When the height of the water level is

10 cm

, the rate (in

{cm}^{2} \sec^{- 1}

) at which the wet conical surface area of the vessel increases is

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 25 June 2022 Shift 2>Q 111

If the angle made by the tangent at the point

(x_{0}, y_{0})

on the curve

x = 12 (t + \sin t \cos t)

y = 12 {(1 + \sin t)}^{2}, 0 < t < \frac{π}{2}

, with the positive

x

-axis is

\frac{π}{3}

, then

y_{0}

is equal to

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 25 June 2022 Shift 2>Q 112

If the line

y = 4 + k x, k > 0

, is the tangent to the parabola

y = x - x^{2}

at the point

P

and

V

is the vertex of the parabola, then the slope of the line through

P

and

V

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 25 June 2022 Shift 2>Q 113

Let

f (x) = |(x - 1) (x^{2} - 2 x - 3)| + x - 3, x \in ℝ

. If

m

and

M

are respectively the number of points of local minimum and local maximum of

f

in the interval

(0, 4)

, then

m + M

is equal to _____.

The sum of absolute maximum and absolute minimum values of the function fx=2x2+3x-2+sinxcosx in the interval 0,1 is

Important Questions on Application of Derivatives

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The sum of absolute maximum and absolute minimum values of the function $f (x) = |2 x^{2} + 3 x - 2| + \sin x \cos x$ in the interval $[0, 1]$ is