Let F : R → R be a thrice differentiable function. Suppose that F 1 = 0 , F 3 = - 4 and F ′ x < 0 for all x ∈ 1 2 , 3 . Let f x = x F ( x ) for all x ∈ R . The correct statement(s) is(are)

f ′ x ≠ 0 for any x ∈ ( 1 , 3 )

f ′ x = 0 for some x ∈ ( 1 , 3 )

Let f : R → R be defined by f x = x 2 - 3 x - 6 x 2 + 2 x + 4 Then which of the following statements is(are) TRUE?

f is increasing in the interval ( 1 , 2 )

Let f : R → R be defined by f x = x 2 - 3 x - 6 x 2 + 2 x + 4 Then which of the following statements is(are) TRUE?

f is increasing in the interval ( 1 , 2 )

Let f : R → R be given by f x = x - 1 x - 2 x - 5 . Define F x = ∫ 0 x f t d t , x > 0 . Then which of the following options is/are correct?

F has two local maxima and one local minimum in 0 , ∞

HARD

JEE Advanced

IMPORTANT

Earn 100

If $f : R \to R$ is a differentiable function such that $f^{'} (x) > 2 f (x)$ for all $x \in R$ and $f (0) = 1$ then

78.57% studentsanswered this correctly

Important Questions on Application of Derivatives

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 24 - Application of Derivatives>JEE Advanced Paper 2 - 2015>Q 5

Let

F : R \to R

be a thrice differentiable function. Suppose that

F (1) = 0, F (3) = - 4

and

F^{'} (x) < 0

for all

x \in (\frac{1}{2}, 3) .

Let

f (x) = x F (x)

for all

x \in R

. The correct statement(s) is(are)

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 24 - Application of Derivatives>JEE Advanced Paper 2 - 2013>Q 6

The function

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

has a local minimum or a local maximum at

x =

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 24 - Application of Derivatives>JEE Advanced Paper 1 - 2021>Q 8

Let $f : R \to R$ be defined by $f (x) = \frac{x^{2} - 3 x - 6}{x^{2} + 2 x + 4}$

Then which of the following statements is(are) TRUE?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 24 - Application of Derivatives>JEE Advanced Paper 1 - 2020>Q 9

Consider all rectangles lying in the region

\{(x, y) \in R \times R : 0 \leq x \leq \frac{π}{2} a n d 0 \leq y \leq 2 \sin (2 x)\}

and having one side on the

x

-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 24 - Application of Derivatives>JEE Advanced Paper 1 - 2019>Q 10

Let

T

denote a curve

y = y (x)

which is in the first quadrant and let the point

(1, 0)

lie on it. Let the tangent to

T

at a point

P

intersect the

y

-axis at

Y_{P} .

P Y_{P}

has length

1

for each point

P

T,

then which of the following options is/are correct?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 24 - Application of Derivatives>JEE Advanced Paper 1 - 2017>Q 11

Let

f (x) = x + \log_{e} x - x \log_{e} x, x \in (0, \infty)

Column 1 contains information about zeros of

f (x)

f' (x)

and

f'' (x)

Column 2 contains information about the limiting behaviour of

f (x)

f' (x)

and

f'' (x)

at infinity.
Column 3 contains information about increasing-decreasing nature of

f (x)

and

f' (x)

Column 1	Column 2	Column 3
(I) $f (x) = 0$ for some $x \in (1, e^{2})$	(i) $\lim_{x \to \infty} f (x) = 0$	(P) $f$ is increasing in $(0, 1)$
(II) $f^{'} (x) = 0$ for some $x in (1, e)$	(ii) $\lim_{x \to \infty} f (x) = - \infty$	(Q) $f$ is decreasing in $(e, e^{2})$
(III) $f^{'} (x) = 0$ for some $x \in (0, 1)$	(iii) $\lim_{x \to \infty} f^{'} (x) = - \infty$	(R) $f^{'}$ is increasing in $(0, 1)$
(IV) $f^{''} (x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x \to \infty} f^{''} (x) = 0$	(S) $f^{'}$ is decreasing in $(e, e^{2})$

Which of the following options is the only CORRECT combination?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 24 - Application of Derivatives>JEE Advanced Paper 1 - 2017>Q 12

Let

f (x) = x + \log_{e} x - x \log_{e} x, x \in (0, \infty) .

•

Column 1 contains information about zeros of

f (x), f^{'} (x)

and

f^{''} (x) .

•

Column 2 contains information about the limiting behaviour of

f (x), f^{'} (x)

and

f^{''} (x)

at infinity.

•

Column 3 contains information about increasing-decreasing nature of

f (x)

and

f^{'} (x) .

Column 1	Column 2	Column 3
(I) $f (x) = 0$ for some $x \in (1, e^{2})$	(i) $\lim_{x \to \infty} f (x) = 0$	(P) $f$ is increasing in (0, 1)
(II) $f^{'} (x) = 0$ for some $x \in (1, e)$	(ii) $\lim_{x \to \infty} f (x) = - \infty$	(Q) $f$ is decreasing in $(e, e^{2})$
(III) $f^{'} (x) = 0$ for some $x \in (0, 1)$	(iii) $\lim_{x \to \infty} f^{'} (x) = - \infty$	(R) $f^{'}$ is increasing in (0, 1)
(IV) $f^{''} (x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x \to \infty} f^{''} (x) = 0$	(S) $f^{'}$ is decreasing in $(e, e^{2})$

Which of the following options is the only Incorrect combination?

HARD

JEE Advanced

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 24 - Application of Derivatives>JEE Advanced Paper 1 - 2017>Q 13

Let

f (x) = x + \log_{e} x - x \log_{e} x, x \in (0, \infty) .

•

Column 1 contains information about zeros of

f (x), f^{'} (x)

and

f^{''} (x) .

•

Column 2 contains information about the limiting behaviour of

f (x), f^{'} (x)

and

f^{''} (x)

at infinity.

•

Column 3 contains information about increasing-decreasing nature of

f (x)

and

f^{'} (x) .

Column 1	Column 2	Column 3
(I) $f (x) = 0$ for some $x \in (1, e^{2})$	(i) $\lim_{x \to \infty} f (x) = 0$	(P) $f$ is increasing in (0, 1)
(II) $f^{'} (x) = 0$ for some $x i n (1, e)$	(ii) $\lim_{x \to \infty} f (x) = - \infty$	(Q) $f$ is decreasing in $(e, e^{2})$
(III) $f^{'} (x) = 0$ for some $x \in (0, 1)$	(iii) $\lim_{x \to \infty} f^{'} (x) = - \infty$	(R) $f^{'}$ is increasing in (0, 1)
(IV) $f^{''} (x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x \to \infty} f^{''} (x) = 0$	(S) $f^{'}$ is decreasing in $(e, e^{2})$

Which of the following options is the only Correct combination?

If f:R→R is a differentiable function such that f′x>2f(x) for all x∈R and f0=1 then

Important Questions on Application of Derivatives

Learn and Prepare for any exam you want !

If $f : R \to R$ is a differentiable function such that $f^{'} (x) > 2 f (x)$ for all $x \in R$ and $f (0) = 1$ then