EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 9th April 2014>Q 101

MEDIUM

JEE Main

IMPORTANT

Earn 100

If $f (x) = {(\frac{3}{5})}^{x} + {(\frac{4}{5})}^{x} - 1, x \in R,$ then the equation $f (x) = 0$ has :

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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 - 6th April 2014>Q 102

If

x = - 1

and

x = 2

are extreme points of

f (x) = α \log |x| + β x^{2} + x

, then

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 24 June 2022 Shift 1>Q 103

For the function

f (x) = 4 \log_{e} (x - 1) - 2 x^{2} + 4 x + 5, x > 1

, which one of the following is NOT correct?

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 24 June 2022 Shift 1>Q 104

If the tangent at the point

(x_{1}, y_{1})

on the curve

y = x^{3} + 3 x^{2} + 5

passes through the origin, then

(x_{1}, y_{1})

does NOT lie on the curve

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 27 - Application of Derivatives>JEE Main - 24 June 2022 Shift 1>Q 105

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

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

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 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?