How many questions from Differentiability appear in JEE?

Differentiability has appeared in 91 of the last 186 JEE Main and JEE Advanced papers — about 49% of them — contributing 105 questions in total across those papers.

Is Differentiability an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 49% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Differentiability questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Differentiability JEE Previous Year Questions with Answers

Q1·MathematicsMultiple correctJEE Advanced 2026

Let

\mathbb{R}

denote the set of all real numbers. Let

f : \mathbb{R} \to \mathbb{R}

be an arbitrary function and let

g : \mathbb{R} \to \mathbb{R}

be the function defined by

g(x) = x f(x)

, for all

x \in \mathbb{R}

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

(A)The function $g$ is always continuous at $x = 0$
(B)If $f$ is continuous at $x = 0$ , then $g$ is differentiable at $x = 0$
(C)If $g$ is differentiable at $x = 0$ , then $f$ is continuous at $x = 0$
(D)If $g$ is differentiable at $x = 0$ , then $\lim_{x \to 0} f(x)$ exists

Q2·MathematicsIntegerJEE Advanced 2026

Consider the function

f : \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \to (-\infty, \infty)

defined by

f(x) = (|x| + |x - 1|) \sin x + [x \sin x]

, where

[x \sin x]

is the greatest integer less than or equal to

x \sin x

. Let

\alpha

be the total number of points in the interval

\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

at which

f

is NOT continuous, and let

\beta

be the total number of points in the interval

\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

at which

f

is NOT differentiable. Then the value of

\alpha + \beta

is ______.

Q3·MathematicsSingle correctJEE Main 2026

For the function

f(x) = e^{\sin|x|} - |x|

,

x \in \mathbb{R}

, consider the following statements: Statement I:

f

is differentiable for all

x \in \mathbb{R}

. Statement II:

f

is increasing in

\left(-\pi, -\frac{\pi}{2}\right)

. In the light of the above statements, choose the correct answer from the options given below:

(A)Both Statement I and Statement II are true
(B)Both Statement I and Statement II are false
(C)Statement I is true but Statement II is false
(D)Statement I is false but Statement II is true

Q4·MathematicsNumericalJEE Main 2026

Let

f(x) = \begin{cases} e^{x-1}, & x < 0 \\ x^{2} - 5x + 6, & x \geq 0 \end{cases}

and

g(x) = f(|x|) + |f(x)|

. If the number of points where

g

is not continuous and is not differentiable are α and β respectively, then

\alpha + \beta

is equal to ______

Q5·MathematicsNumericalJEE Main 2026

The number of points, at which the function

f(x) = \max\{6x, 2 + 3x^2\} + |x - 1|\left|\cos\left|x^2 - \frac{1}{4}\right|\right|

,

x \in (-\pi, \pi)

, is not differentiable, is _______.

Q6·MathematicsSingle correctJEE Main 2026

Let α, β ∈

\mathbb{R}

be such that the function

f(\mathrm{x}) = \begin{cases} 2\alpha(\mathrm{x}^{2} - 2) + 2\beta \mathrm{x} & ,\mathrm{x} < 1 \\ (\alpha + 3)\mathrm{x} + (\alpha - \beta) & ,\mathrm{x} \geq 1 \end{cases}

be differentiable at all x∈

\mathbb{R}

. Then 34(α + β) is equal to

(A)84
(B)48
(C)36
(D)24

Q7·MathematicsSingle correctJEE Main 2026

Let [•] denote the greatest integer function, and let

f(x) = \min\left\{\sqrt{2}x, x^2\right\}

. Let S = {x ∈ (–2, 2) : the function

g(x) = |x|[x^2]

is discontinuous at x}. Then

\sum_{x \in S} f(x)

equals :

(A) $2 - \sqrt{2}$
(B) $2\sqrt{6} - 3\sqrt{2}$
(C) $1 - \sqrt{2}$
(D) $\sqrt{6} - 2\sqrt{2}$

Q8·MathematicsSingle correctJEE Main 2026

Let

f(x) = x^3 + x^2 f'(1) + 2x f''(2) + f'''(3)

, x ∈ R. Then the value of

f'(5)

is :

(A) $\frac{62}{5}$
(B) $\frac{657}{5}$
(C) $\frac{2}{5}$
(D) $\frac{117}{5}$

Q9·MathematicsNumericalJEE Advanced 2025

Let R denote the set of all real numbers. Let

f : R \rightarrow R

and

g : R \rightarrow (0, 4)

be functions defined by

f(x) = \log_e\left(x^2 + 2x + 4\right)

, and

g(x) = \frac{4}{1 + e^{-2x}}

. Define the composite function

f \circ g^{-1}

by

\left(f \circ g^{-1}\right)(x) = f\left(g^{-1}(x)\right)

, where

g^{-1}

is the inverse of the function g. Then the value of the derivative of the composite function

f \circ g^{-1}

at x = 2 is ______

Q10·MathematicsSingle correctJEE Main 2025

Let the function

f(x)=(x^2-1)|x^2-ax+2|+\cos|x|

be not differentiable at the two points

x=\alpha=2

and

x=\beta

. Then the distance of the point

(\alpha,\beta)

from the line

12x+5y+10=0

is equal to:

(A)3
(B)4
(C)2
(D)5

Q11·MathematicsSingle correctJEE Main 2025

Let

f

be differentiable with

f(0)=1

and

f(x+y)=f(x)f(y)

. Under the given conditions,

\displaystyle\sum_{n=1}^{100}f'(n)

equals:

(A)2384
(B)2525
(C)5220
(D)2406

Q12·MathematicsIntegerJEE Main 2025

Let

f(x)=\begin{cases}-3ax^2-2,&x<1\\ a^2+bx,&x\ge1\end{cases}

be differentiable for all

x\in\mathbb{R}

(

a>1

). With the region enclosed by

y=f(x)

and the given line equal to

\beta\sqrt3

(

\alpha,\beta\in\mathbb{Z}

), find

\alpha+\beta

.

Q13·MathematicsSingle correctJEE Advanced 2024

Let

f : R \to R

and

g : R \to R

be functions defined by

f(x) = \begin{cases} x|x|\sin\left(\frac{1}{x}\right), & x \neq 0, \\ 0, & x = 0, \end{cases}

and

g(x) = \begin{cases} 1 - 2x, & 0 \leq x \leq \frac{1}{2}, \\ 0, & \text{otherwise.} \end{cases}

Let

a, b, c, d \in R

. Define the function

h : R \to R

by

h(x) = af(x) + b\left(g(x) + g\left(\frac{1}{2} - x\right)\right) + c(x - g(x)) + d\,g(x), x \in R

Match each entry in List-I to the correct entries in List-II. The correct option is:

List-I	List-II
P.If $a = 0$ , $b = 1$ , $c = 0$ and $d = 0$ , then	1. $h$ is one-one.
Q.If $a = 1$ , $b = 0$ , $c = 0$ and $d = 0$ , then	2. $h$ is onto.
R.If $a = 0$ , $b = 0$ , $c = 1$ and $d = 0$ , then	3. $h$ is differentiable on $R$ .
S.If $a = 0$ , $b = 0$ , $c = 0$ and $d = 1$ , then	4.the range of $h$ is $[0, 1]$ .
	5.the range of $h$ is $\{0, 1\}$ .

(A)(P) $\to$ (4) (Q) $\to$ (3) (R) $\to$ (1) (S) $\to$ (2)
(B)(P) $\to$ (5) (Q) $\to$ (2) (R) $\to$ (4) (S) $\to$ (3)
(C)(P) $\to$ (5) (Q) $\to$ (3) (R) $\to$ (2) (S) $\to$ (4)
(D)(P) $\to$ (4) (Q) $\to$ (2) (R) $\to$ (1) (S) $\to$ (3)

Q14·MathematicsSingle correctJEE Main 2024

If

\log_{e}y=3\sin^{-1}x

, then

(1-x^{2})y''-xy'

at

x=\dfrac{1}{2}

is equal to:

(A) $9e^{5\pi/6}$
(B) $3e^{5\pi/6}$
(C) $3e^{\pi/2}$
(D) $9e^{\pi/2}$

Q15·MathematicsSingle correctJEE Main 2024

Let

f(x) = ax^3 + bx^2 + cx + 41

be such that

f(1) = 40

,

f'(1) = 2

and

f''(1) = 4

. Then

a^2 + b^2 + c^2

is equal to:

(A)62
(B)73
(C)54
(D)51

Q16·MathematicsSingle correctJEE Main 2024

Let

f:(-\infty,\infty)-\{0\}\to\mathbb{R}

be a differentiable function such that

f'(1)=\displaystyle\lim_{a\to\infty}a^{2}f\left(\tfrac{1}{a}\right)

. Then

\displaystyle\lim_{a\to\infty}\frac{a(a+1)}{2}\tan^{-1}\left(\tfrac{1}{a}\right)+a^{2}-2\log_{e}a

is equal to

(A) $\tfrac{3}{2}+\tfrac{\pi}{4}$
(B) $\tfrac{3}{8}+\tfrac{\pi}{4}$
(C) $\tfrac{5}{2}+\tfrac{\pi}{8}$
(D) $\tfrac{3}{4}+\tfrac{\pi}{8}$

Q17·MathematicsSingle correctJEE Main 2024

Suppose for a differentiable function h,

h(0) = 0

,

h(1) = 1

and

h'(0) = h'(1) = 2

. If

g(x) = h(e^x)\,e^{h(x)}

, then

g'(0)

is equal to:

(A)5
(B)3
(C)8
(D)4

Q18·MathematicsSingle correctJEE Main 2024

If

f(x)=x^{3}\sin\left(\tfrac{1}{x}\right)

for

x\neq 0

and

f(0)=0

, then

(A) $f''(0)=1$
(B) $f''\left(\tfrac{2}{\pi}\right)=\tfrac{24-\pi^{2}}{2\pi}$
(C) $f''\left(\tfrac{2}{\pi}\right)=\tfrac{12-\pi^{2}}{2\pi}$
(D) $f''(0)=0$

Q19·MathematicsSingle correctJEE Main 2024

Let

f(x) = x^5 + 2x^3 + 3x + 1

,

x \in \mathbb{R}

, and

g(x)

be a function such that

g(f(x)) = x

for all

x \in \mathbb{R}

. Then

\dfrac{g(7)}{g'(7)}

is equal to:

(A) $7$
(B) $42$
(C) $1$
(D) $14$

Q20·MathematicsSingle correctJEE Main 2024

If

y(\theta)=\dfrac{2\cos\theta+\cos 2\theta}{\cos 3\theta+4\cos 2\theta+5\cos\theta+2}

, then at

\theta=\dfrac{\pi}{2}

,

y''+y'+y

is equal to:

(A) $\dfrac{\pi}{2}$
(B)1
(C) $\dfrac{1}{2}$
(D)2

Q21·MathematicsNumericalJEE Main 2024

Let

f:\mathbb{R}\to\mathbb{R}

be a thrice differentiable function such that

f(0)=0

,

f(1)=1

,

f(2)=-1

,

f(3)=2

and

f(4)=-2

. Then, the minimum number of zeros of

\left(3f'f''+ff'''\right)(x)

is

Q22·MathematicsSingle correctJEE Main 2024

Let

f(x)=x^5+2e^{x/4}

for all

x\in\mathbb{R}

. Consider a function

g(x)

such that

(g\circ f)(x)=x

for all

x\in\mathbb{R}

. Then the value of

8g'(2)

is:

(A) $16$
(B) $4$
(C) $8$
(D) $2$

Q23·MathematicsSingle correctJEE Main 2024

Let

a

and

b

be real constants such that the function

f

defined by

f(x)=\begin{cases}x^2+3x+a, & x\le 1\\ bx+2, & x>1\end{cases}

be differentiable on

R

. Then the value of

\displaystyle\int_{-2}^{2}f(x)\,dx

equals:

(A) $\dfrac{15}{6}$
(B) $\dfrac{19}{6}$
(C) $21$
(D) $17$

Q24·MathematicsSingle correctJEE Main 2024

Let

g:\mathbb{R}\to\mathbb{R}

be a non constant twice differentiable function such that

g'\left(\dfrac12\right)=g'\left(\dfrac32\right)

. If a real valued function

f

is defined as

f(x)=\dfrac12[g(x)+g(2-x)]

, then:

(A) $f'(x)=0$ for atleast two $x$ in $(0,2)$
(B) $f'(x)=0$ for exactly one $x$ in $(0,1)$
(C) $f'(x)=0$ for no $x$ in $(0,1)$
(D) $f'\left(\dfrac32\right)+f'\left(\dfrac12\right)=1$

Q25·MathematicsNumericalJEE Main 2024

If the function

f(x)=\begin{cases}\dfrac{1}{|x|} & ,\ |x|\ge2\\ ax^2+2b & ,\ |x|<2\end{cases}

is differentiable on

\mathbb{R}

, then

48(a+b)

is equal to ___

Q26·MathematicsNumericalJEE Main 2024

Let

f(x)=2^x-x^2

,

x\in R

. If m and n are respectively the number of points at which the curves

y=f(x)

and

y=f'(x)

intersects the x-axis, then the value of

m+n

is ______.

Q27·MathematicsMultiple correctJEE Advanced 2023

Let

f : (0, 1) \to \mathbb{R}

be the function defined as

f(x) = [4x]\left(x - \frac{1}{4}\right)^{2}\left(x - \frac{1}{2}\right)

, where

[x]

denotes the greatest integer less than or equal to x . Then which of the following statements is(are) true?

(A)The function $f$ is discontinuous exactly at one point in $(0, 1)$
(B)There is exactly one point in $(0, 1)$ at which the function $f$ is continuous but NOT differentiable
(C)The function f is NOT differentiable at more than three points in $(0, 1)$
(D)The minimum value of the function f is $-\frac{1}{512}$

Q28·MathematicsSingle correctJEE Main 2023

For the differentiable function

f:\mathbb{R}\setminus\{0\}\to\mathbb{R}

, if

3f(x)+2f\left(\dfrac{1}{x}\right)=\dfrac{1}{x}-10

, then

\left|f(3)+f\,'\!\left(\dfrac14\right)\right|

is equal to:

(A) $7$
(B) $\dfrac{33}{5}$
(C) $\dfrac{29}{5}$
(D) $13$

Q29·MathematicsSingle correctJEE Main 2023

Let

f(x) = x^2 - x + |-x + [x]|

, where

x \in \mathbb{R}

and

[t]

denotes the greatest integer less than or equal to

t

. Then,

f

is

(A)continuous at $x = 0$ , but not continuous at $x = 1$
(B)continuous at $x = 0$ and $x = 1$
(C)not continuous at $x = 0$ and $x = 1$
(D)continuous at $x = 1$ , but not continuous at $x = 0$

Q30·MathematicsNumericalJEE Main 2023

Let k and m be positive real numbers such that the function

f(x)=\begin{cases}3x^{2}+k\sqrt{x+1},&0<x<1\\mx^{2}+k^{2},&x\geq1\end{cases}

is differentiable for all

x>0

. Then

\dfrac{8f'(8)}{f'\left(\dfrac{1}{8}\right)}

is equal to

Q31·MathematicsSingle correctJEE Main 2023

If

2^{xy}+3^{yx}=20

, then

\dfrac{dy}{dx}

at

(2,2)

is equal to:

(A) $-\left(\dfrac{3+\log_e 8}{2+\log_e 4}\right)$
(B) $-\left(\dfrac{2+\log_e 8}{3+\log_e 4}\right)$
(C) $-\left(\dfrac{3+\log_e 16}{4+\log_e 8}\right)$
(D) $-\left(\dfrac{3+\log_e 4}{2+\log_e 8}\right)$

Q32·MathematicsSingle correctJEE Main 2023

If

y(x)=x^x,\ x>0

, then

y''(2)-2y'(2)

is equal to

(A) $4\log_e 2+2$
(B) $8\log_e 2-2$
(C) $4(\log_e 2)^2+2$
(D) $4(\log_e 2)^2-2$

Q33·MathematicsSingle correctJEE Main 2023

Let

y=(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})

. Then

y'-y''

at

x=-1

is equal to:

(A) $976$
(B) $944$
(C) $464$
(D) $496$

Q34·MathematicsSingle correctJEE Main 2023

If

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

then

(A) $f(1)+f(2)+f(3)=f(0)$
(B) $2f(0)-f(1)+f(3)=f(2)$
(C) $3f(1)+f(2)=f(3)$
(D) $f(3)-f(2)=f(1)$

Q35·MathematicsSingle correctJEE Main 2023

Let

f(x)=\begin{cases}x^2\sin\!\left(\tfrac{1}{x}\right), & x\ne 0\\ 0, & x=0\end{cases}

. Then at

x=0

(A) $f$ is continuous but not differentiable
(B) $f$ and $f'$ both are continuous
(C) $f'$ is continuous but not differentiable
(D) $f$ is continuous but $f'$ is not continuous

Q36·MathematicsNumericalJEE Main 2022

If [t] denotes the greatest integer

\le t

, then number of points, at which the function

f(x) = 4|2x + 3| + 9\left[x + \frac{1}{2}\right] - 12[x + 20]

is not differentiable in the open interval

(-20, 20)

, is___.

Q37·MathematicsSingle correctJEE Main 2022

The number of points, where the function

f : \mathbf{R} \rightarrow \mathbf{R}

,

f(x) = |x-1|\cos|x-2|\sin|x-1| + (x-3)\left|x^{2}-5x+4\right|

, is NOT differentiable, is :

(A)1
(B)2
(C)3
(D)4

Q38·MathematicsSingle correctJEE Main 2022

Let

x(t)=2\sqrt{2}\cos t\sqrt{\sin 2t}

and

y(t)=2\sqrt{2}\sin t\sqrt{\sin 2t}

,

t \in \left(0,\frac{\pi}{2}\right)

. Then

\frac{1+\left(\frac{dy}{dx}\right)^{2}}{\frac{d^{2}y}{dx^{2}}}

at

t=\frac{\pi}{4}

is equal to

(A) $\frac{-2\sqrt{2}}{3}$
(B) $\frac{2}{3}$
(C) $\frac{1}{3}$
(D) $\frac{-2}{3}$

Q39·MathematicsNumericalJEE Main 2022

For the curve

C : (x^{2} + y^{2} - 3) + (x^{2} - y^{2} - 1)^{5} = 0

, the value of

3y' - y^{3}y''

, at the point

(\alpha, \alpha)

,

\alpha > 0

, on C, is equal to ________.

Q40·MathematicsSingle correctJEE Main 2022

Let

f(x) = 2 + |x| - |x-1| + |x+1|

,

x \in \mathbf{R}

. Consider

(S1) : f'\left(-\frac{3}{2}\right) + f'\left(-\frac{1}{2}\right) + f'\left(\frac{1}{2}\right) + f'\left(\frac{3}{2}\right) = 2

(S2) : \int_{-2}^{2} f(x)dx = 12

Then,

(A)both (S1) and (S2) are correct
(B)both (S1) and (S2) are wrong
(C)only (S1) is correct
(D)only (S2) is correct

Q41·MathematicsSingle correctJEE Main 2022

The value of

\log_{e}2\,\frac{d}{dx}\left(\log_{\cos x}\operatorname{cosec}x\right)

at

x=\frac{\pi}{4}

is

(A) $-2\sqrt{2}$
(B) $2\sqrt{2}$
(C)-4
(D)4

Q42·MathematicsNumericalJEE Main 2022

Let

f

and g be twice differentiable even functions on

(-2, 2)

such that

f\left(\frac{1}{4}\right) = 0, f\left(\frac{1}{2}\right) = 0, f(1) = 1

and

g\left(\frac{3}{4}\right) = 0, g(1) = 2

Then, the minimum number of solutions of

f(x)\ g''(x) + f'(x)g'(x) = 0

in

(-2,2)

is equal to____.

Q43·MathematicsSingle correctJEE Main 2022

Let

f : R \to R

be a function defined by :

f(x) = \begin{cases} \max\limits_{t \le x}\{t^{3} - 3t\}\ ; & x \le 2 \\ x^{2} + 2x - 6\ ; & 2 < x < 3 \\ [x - 3] + 9\ ; & 3 \le x \le 5 \\ 2x + 1\ ; & x > 5 \end{cases}

Where [t] is the greatest integer less than or equal to t. Let m be the number of points where f is not differentiable and

I = \int_{-2}^{2} f(x)dx

. Then the ordered pair (m, I) is equal to :

(A) $\left(3, \frac{27}{4}\right)$
(B) $\left(3, \frac{23}{4}\right)$
(C) $\left(4, \frac{27}{4}\right)$
(D) $\left(4, \frac{23}{4}\right)$

Q44·MathematicsSingle correctJEE Main 2022

If

\cos^{-1}\left(\frac{y}{2}\right) = \log_{e}\left(\frac{x}{5}\right)^{5}, |y| < 2

, then :

(A) $x^{2}y'' + xy' - 25y = 0$
(B) $x^{2}y'' - xy' - 25y = 0$
(C) $x^{2}y'' - xy' + 25y = 0$
(D) $x^{2}y'' + xy' + 25y = 0$

Q45·MathematicsSingle correctJEE Main 2022

Let

f : R \to R

be defined as

f(x) = x^3 + x - 5

. If

g(x)

is a function such that

f(g(x)) = x

, ∀

x \in R

, then

g'(63)

is equal to ______.

(A) $\frac{1}{49}$
(B) $\frac{3}{49}$
(C) $\frac{43}{49}$
(D) $\frac{91}{49}$

Q46·MathematicsSingle correctJEE Main 2022

If

y = \tan^{-1}(\sec x^3 - \tan x^3)

.

\dfrac{\pi}{2} < x^3 < \dfrac{3\pi}{2}

, then

(A) $xy'' + 2y' = 0$
(B) $x^2 y'' - 6y + \dfrac{3\pi}{2} = 0$
(C) $x^2 y'' - 6y + 3\pi = 0$
(D) $xy'' - 4y' = 0$

Q47·MathematicsSingle correctJEE Main 2021

Let

f

be any continuous function on

[0, 2]

and twice differentiable on

(0, 2)

. If

f(0) = 0

,

f(1) = 1

and

f(2) = 2

, then

(A) $f''(x) = 0$ for all $x \in (0, 2)$
(B) $f''(x) = 0$ for some $x \in (0, 2)$
(C) $f'(x) = 0$ for some $x \in [0, 2]$
(D) $f''(x) > 0$ for all $x \in (0, 2)$

Q48·MathematicsSingle correctJEE Main 2021

The function

f(x) = \left|x^{2} - 2x - 3\right| \cdot e^{\left|9x^{2} - 12x + 4\right|}

is not differentiable at exactly :

(A)four points
(B)three points
(C)two points
(D)one point

Q49·MathematicsNumericalJEE Main 2021

If

y^{1/4} + y^{-1/4} = 2x

, and

\left(x^{2} - 1\right)\frac{d^{2}y}{dx^{2}} + \alpha x\frac{dy}{dx} + \beta y = 0

, then

|\alpha - \beta|

is equal to ___________.

Q50·MathematicsSingle correctJEE Main 2021

Let [t] denote the greatest integer less than or equal to t. Let

f(x) = x - [x]

,

g(x) = 1 - x + [x]

, and

h(x) = \min\{f(x), g(x)\}

,

x \in [-2, 2]

. Then h is :

(A)continuous in $[-2, 2]$ but not differentiable at more than four points in $(-2, 2)$
(B)not continuous at exactly three points in $[-2, 2]$
(C)continuous in $[-2, 2]$ but not differentiable at exactly three points in $(-2, 2)$
(D)not continuous at exactly four points in $[-2, 2]$

Q51·MathematicsNumericalJEE Main 2021

If

y = y(x)

is an implicit function of x such that

\log_{e}(x + y) = 4xy

, then

\frac{d^{2}y}{dx^{2}}

at

x = 0

is equal to ________.

Q52·MathematicsSingle correctJEE Main 2021

Let f : [0, ∞) → [0, 3] be a function defined by f(x) =

\begin{cases} \max\left\{ \sin t : 0 \le t \le x \right\}, & 0 \le x \le \pi \\ 2 + \cos x, & x > \pi \end{cases}

Then which of the following is true?

(A)f is differentiable everywhere in $\left( 0, \infty \right)$
(B)f is not continuous exactly at two points in $\left( 0, \infty \right)$
(C)f is continuous everywhere but not differentiable exactly at two points in $\left( 0, \infty \right)$
(D)f is continuous everywhere but not differentiable exactly at one point in $\left( 0, \infty \right)$

Q53·MathematicsSingle correctJEE Main 2021

Let

f:[0,\infty) \rightarrow [0,\infty)

be defined as

f(x)=\int_{0}^{x}[y]\,dy

where [x] is the greatest integer less than or equal to x. Which of the following is true?

(A)f is differentiable at every point in $[0,\infty)$
(B)f is continuous everywhere except at the integer points in $[0,\infty)$ .
(C)f is continuous at every point in $[0,\infty)$ and differentiable except at the integer points.
(D)f is both continuous and differentiable except at the integer points in $[0,\infty)$ .

Q54·MathematicsSingle correctJEE Main 2021

If

f\left(x\right) = \begin{cases} \int\limits_{0}^{x} \left(5 + \left|1 - t\right|\right)dt, & x > 2 \\ 5x + 1, & x \leq 2 \end{cases}

, then

(A)f(x) is not continuous at x = 2
(B)f(x) is not differentiable at x = 1
(C)f(x) is continuous but not differentiable at x = 2
(D)f(x) is everywhere differentiable

Q55·MathematicsNumericalJEE Main 2021

Let

f : R \rightarrow R

be a function defined as

f(x) = \begin{cases} 3\left(1-\dfrac{|x|}{2}\right) & \text{if } |x| \leq 2 \\ 0 & \text{if } |x| > 2 \end{cases}

Let

g : R \rightarrow R

be given by

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

. If n and m denote the number of points in R where g is the continuous and not differentiable, respectively, then n + m is equal to............

Q56·MathematicsNumericalJEE Main 2021

Let a function

g:[0,4]\to R

be defined as

g(x)=\begin{cases}\max\limits_{0\le t\le x}\{t^{3}-6t^{2}+9t-3\}, & 0\le x\le 3 \\ 4-x, & 3<x\le 4\end{cases}

, then the number of points in the interval

(0,4)

where

g(x)

is NOT differentiable, is…….

Q57·MathematicsSingle correctJEE Main 2021

If

f\left(x\right) = \begin{cases} \dfrac{1}{\left|x\right|} & ; \;\; \left|x\right| \geq 1 \\ ax^{2} + b & ; \;\; \left|x\right| < 1 \end{cases}

is differentiable at every point of the domain, then the values of a and b are respectively :

(A) $\dfrac{1}{2}, \dfrac{1}{2}$
(B) $\dfrac{1}{2}, -\dfrac{3}{2}$
(C) $\dfrac{5}{2}, -\dfrac{3}{2}$
(D) $-\dfrac{1}{2}, \dfrac{3}{2}$

Q58·MathematicsNumericalJEE Main 2021

Let

f

: R → R satisfy the equation

f(x + y) = f(x) \cdot f(y)

for all x, y ∈ R and f(x) ≠ 0 for any x ∈ R. If the function

f

is differentiable at x = 0 and

f'(0) = 3

, then

\lim_{h \to 0} \frac{1}{h}\left(f(h) - 1\right)

is equal to ________ .

Q59·MathematicsNumericalJEE Main 2021

If

f(x) = \sin\left(\cos^{-1}\left(\frac{1 - 2^{2x}}{1 + 2^{2x}}\right)\right)

and its first derivative with respect to x is

-\frac{b}{a}\log_e 2

when

x = 1

, where a and b are integers, then the minimum value of

|a^2 - b^2|

is _______ .

Q60·MathematicsSingle correctJEE Main 2021

Let

f : S \rightarrow S

where

S = (0, \infty)

be a twice differentiable function such that

f(x + 1) = x f(x)

. If

g : S \rightarrow R

be defined as

g(x)=\log_{e}f(x)

, then the value of

\left|g''(5)-g''(1)\right|

is equal to :

(A) $\frac{205}{144}$
(B) $\frac{197}{144}$
(C) $\frac{187}{144}$
(D) $1$

Q61·MathematicsSingle correctJEE Main 2021

Let the functions f : ℝ → ℝ and g : ℝ → ℝ be defined as :

f(x) = \begin{cases} x + 2, & x < 0 \\ x^{2}, & x \ge 0 \end{cases}

and

g(x) = \begin{cases} x^{3}, & x < 1 \\ 3x - 2, & x \ge 1 \end{cases}

Then, the number of points in ℝ where (fog)(x) is NOT differentiable is equal to :

(A)3
(B)1
(C)0
(D)2

Q62·MathematicsSingle correctJEE Main 2021

Let f be any function defined on R and let it satisfy the condition :

(\left|f(x) - f(y)\right| \le \left|(x - y)^{2}\right|, \forall (x, y) \in R

If f(0) = 1, then:

(A)f(x) < 0, ∀x ∈ R
(B)f(x) can take any value in R
(C)f(x) = 0, ∀ x ∈ R
(D)f(x) >0, ∀ x ∈ R

Q63·MathematicsNumericalJEE Main 2021

A function f is defined on [–3,3] as f ()x={min {| x |,2−x

^{2}

}, −2≤x≤2 [| x |],2<| x |≤3 where [x] denotes the greatest integer ≤ x. The number of points, where f is not differentiable in (–3,3) is ______.

Q64·MathematicsNumericalJEE Main 2021

The number of points, at which the function

f(x) = |2x + 1| - 3|x + 2| + |x^2 + x - 2|

,

x \in R

is not differentiable, is __________.

Q65·MathematicsMultiple correctJEE Advanced 2020

Let the function

f : \mathbb{R} \to \mathbb{R}

be defined by

f(x) = x^{3} - x^{2} + (x - 1)\sin x

and let

g : \mathbb{R} \to \mathbb{R}

be an arbitrary function. Let fg

: \mathbb{R} \to \mathbb{R}

be the product function defined by

(f g)(x) = f(x)\, g(x)

. Then which of the following statements is/are TRUE ?

(A)If g is continuous at $x = 1$ , then fg is differentiable at $x = 1$
(B)If fg is differentiable at $x = 1$ , then g is continuous at $x = 1$
(C)If g is differentiable at $x = 1$ , then fg is differentiable at $x = 1$
(D)If fg is differentiable at $x = 1$ , then g is differentiable at $x = 1$

Q66·MathematicsIntegerJEE Advanced 2020

Let the functions

f : (-1,1) \to \mathbb{R}

and

g : (-1,1) \to (-1,1)

be defined by

f(x) = |2x-1| + |2x+1|

and

g(x) = x - [x]

, where

[x]

denotes the greatest integer less than or equal to

x

. Let

f \circ g : (-1,1) \to \mathbb{R}

be the composite function defined by

(f \circ g)(x) = f(g(x))

. Suppose

c

is the number of points in the interval

(-1,1)

at which

f \circ g

is NOT continuous, and suppose

d

is the number of points in the interval

(-1,1)

at which

f \circ g

is NOT differentiable. Then the value of

c + d

is ________

Q67·MathematicsMultiple correctJEE Advanced 2020

Let

f : \mathbb{R} \to \mathbb{R}

and

g : \mathbb{R} \to \mathbb{R}

be functions satisfying

f(x+y) = f(x) + f(y) + f(x)f(y)

and

f(x) = x g(x)

for all

x, y \in \mathbb{R}

. If

\lim_{x \to 0} g(x) = 1

, then which of the following statements is/are TRUE?

(A) $f$ is differentiable at every $x \in \mathbb{R}$
(B)If $g(0) = 1$ , then $g$ is differentiable at every $x \in \mathbb{R}$
(C)The derivative $f'(1)$ is equal to 1
(D)The derivative $f'(0)$ is equal to 1

Q68·MathematicsSingle correctJEE Main 2020

Let

f : R \to R

be a function defined by

f(x) = \max\{x, x^{2}\}

. Let S denote the set of all points in R, where f is not differentiable. Then:

(A) $\{0, 1\}$
(B) $\{0\}$
(C) $\phi$ (an empty set)
(D) $\{1\}$

Q69·MathematicsNumericalJEE Main 2020

Let

f:R\to R

be defined as

f(x)=\begin{cases}x^{5}\sin\left(\dfrac{1}{x}\right)+5x^{2}, & x<0\\ 0, & x=0\\ x^{5}\cos\left(\dfrac{1}{x}\right)+\lambda x^{2}, & x>0\end{cases}

. The value of

\lambda

for which

f''(0)

exists, is ____.

Q70·MathematicsSingle correctJEE Main 2020

The derivation of

\tan^{-1}\left(\dfrac{\sqrt{1+x^2}-1}{x}\right)

with respect to

\tan^{-1}\left(\dfrac{2x\sqrt{1-x^2}}{1-2x^2}\right)

at

x=\dfrac{1}{2}

is:

(A) $\dfrac{2\sqrt{3}}{3}$
(B) $\dfrac{\sqrt{3}}{10}$
(C) $\dfrac{\sqrt{3}}{12}$
(D) $\dfrac{2\sqrt{3}}{5}$

Q71·MathematicsSingle correctJEE Main 2020

If the function

f(x)=\begin{cases} k_{1}(x-\pi)^{2}-1 & x \le \pi \\ k_{2}\cos x & x > \pi \end{cases}

is twice differentiable, then the ordered pair

(k_{1},k_{2})

is equal to:

(A) $(1,0)$
(B) $(1,1)$
(C) $\left(\dfrac{1}{2},-1\right)$
(D) $\left(\dfrac{1}{2},1\right)$

Q72·MathematicsSingle correctJEE Main 2020

If

\left( a + \sqrt{2}\, b \cos x \right)\left( a - \sqrt{2}\, b \cos y \right) = a^{2} - b^{2}

, where

a > b > 0

, then

\frac{dx}{dy}

at

\left( \frac{\pi}{4}, \frac{\pi}{4} \right)

is:

(A) $\frac{a - 2b}{a + 2b}$
(B) $\frac{2a + b}{2a - 2}$
(C) $\frac{a + b}{a - b}$
(D) $\frac{a - b}{a + b}$

Q73·MathematicsNumericalJEE Main 2020

Suppose a differentiable function

f(x)

satisfies the identity

f(x+y) = f(x) + f(y) + xy^{2} + x^{2}y

, for all real x and y. If

\lim_{x \to 0} \frac{f(x)}{x} = 1

then

f'(3)

is equal to __________.

Q74·MathematicsSingle correctJEE Main 2020

The function

f(x) = \begin{cases} \frac{\pi}{4} + \tan^{-1} x, & |x| \leq 1 \\ \frac{1}{2}\left(|x| - 1\right), & |x| > 1 \end{cases}

is:

(A)both continuous and differentiable on R − {1}.
(B)both continuous and differentiable on R − {−1}.
(C)continuous on R − {1} and differentiable on R − {−1, 1}.
(D)continuous on R − {−1} and differentiable on R − {−1, 1}.

Q75·MathematicsSingle correctJEE Main 2020

If

y^{2}+\log_{e}\left(\cos^{2}x\right)=y

,

x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)

, Then:

(A) $\left|y'(0)\right|+\left|y''(0)\right|=3$
(B) $\left|y''(0)\right|=2$
(C) $\left|y'(0)\right|+\left|y''(0)\right|=1$
(D) $\left|y''(0)\right|=0$

Q76·MathematicsSingle correctJEE Main 2020

Let f and g be differentiable functions on R such that fog is the identity function. If for some

a, b \in R, g'(a) = 5

and

g(a) = b

, then

f'(b)

is equal to:

(A) $1$
(B) $5$
(C) $\frac{1}{5}$
(D) $\frac{2}{5}$

Q77·MathematicsSingle correctJEE Main 2020

Let

f(x)=x\cos^{-1}(-\sin|x|), x\in\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]

, then which of the following is true?

(A) $f'(0)=-\dfrac{\pi}{2}$
(B)f is not differentiable at x = 0
(C)f' is decreasing in $\left(-\dfrac{\pi}{2},0\right)$ and increasing in $\left(0,\dfrac{\pi}{2}\right)$
(D)f' is increasing in $\left(-\dfrac{\pi}{2},0\right)$ and decreasing in $\left(0,\dfrac{\pi}{2}\right)$

Q78·MathematicsSingle correctJEE Main 2020

Let

y=y(x)

be a function of x satisfying

y\sqrt{1-x^{2}}=k-x\sqrt{1-y^{2}}

where k is a constant and

y\left(\dfrac{1}{2}\right)=-\dfrac{1}{4}

. Then

\dfrac{dy}{dx}

at

x=\dfrac{1}{2}

, is equal to:

(A) $-\dfrac{\sqrt{5}}{4}$
(B) $-\dfrac{\sqrt{5}}{2}$
(C) $\dfrac{2}{\sqrt{5}}$
(D) $\dfrac{\sqrt{5}}{2}$

Q79·MathematicsNumericalJEE Main 2020

Let S be the set of points where the function,

f(x) = \left|2 - |x-3|\right|, x \in R

, is not differentiable. Then

\sum\limits_{x \in S} f(f(x))

is equal to _________.

Q80·MathematicsSingle correctJEE Main 2020

If

y(\alpha)=\sqrt{2\left(\frac{\tan\alpha+\cot\alpha}{1+\tan^{2}\alpha}\right)+\frac{1}{\sin^{2}\alpha}}

,

\alpha\in\left(\frac{3\pi}{4},\pi\right)

then

\frac{dy}{d\alpha}

at

\alpha=\frac{5\pi}{6}

is:

(A)4
(B) $-\frac{1}{4}$
(C)−4
(D) $\frac{4}{3}$

Q81·MathematicsMultiple correctJEE Advanced 2019

Let

f : R \to R

by given by

f(x) = \begin{cases} x^5 + 5x^4 + 10x^3 + 10x^2 + 3x + 1, & x < 0; \\ x^2 - x + 1, & 0 \le x < 1; \\ \frac{2}{3}x^3 - 4x^2 + 7x - \frac{8}{3}, & 1 \le x < 3; \\ (x - 2)\log_e (x - 2) - x + \frac{10}{3}, & x \ge 3. \end{cases}

Then which of the following options is/are correct?

(A) $f'$ has a local maximum at $x = 1$
(B) $f'$ is NOT differentiable at $x = 1$
(C)f is onto
(D)f is increasing on $(-\infty, 0)$

Q82·MathematicsMultiple correctJEE Advanced 2019

Let

f : \mathbb{R} \to \mathbb{R}

be a function. We say that f has PROPERTY 1 if

\lim\limits_{h \to 0} \dfrac{f(h) - f(0)}{\sqrt{|h|}}

exists and is finite, and PROPERTY 2 if

\lim\limits_{h \to 0} \dfrac{f(h) - f(0)}{h^{2}}

exists and if finite. Then which of the following options is/are correct?

(A) $f(x) = x|x|$ has PROPERTY 2
(B) $f(x) = \sin x$ has PROPERTY 2
(C) $f(x) = |x|$ has PROPERTY 1
(D) $f(x) = x^{2/3}$ has PROPERTY 1

Q83·MathematicsSingle correctJEE Main 2019

The derivative of

\tan^{-1}\left(\dfrac{\sin x - \cos x}{\sin x + \cos x}\right)

, with respect to

\dfrac{x}{2}

, where

\left(x \in \left(0,\dfrac{\pi}{2}\right)\right)

is:

(A)2
(B) $\dfrac{1}{2}$
(C)1
(D) $\dfrac{2}{3}$

Q84·MathematicsSingle correctJEE Main 2019

If

e^{y} + xy = e

, the ordered pair

\left(\dfrac{dy}{dx}, \dfrac{d^{2}y}{dx^{2}}\right)

at x = 0 is equal to :

(A) $\left(\dfrac{1}{e}, -\dfrac{1}{e^{2}}\right)$
(B) $\left(\dfrac{1}{e}, \dfrac{1}{e^{2}}\right)$
(C) $\left(-\dfrac{1}{e}, \dfrac{1}{e^{2}}\right)$
(D) $\left(-\dfrac{1}{e}, -\dfrac{1}{e^{2}}\right)$

Q85·MathematicsSingle correctJEE Main 2019

Let f(x) = log

_e

(sinx), (0 < x < π) and g(x) =

\sin^{-1}(e^{-x})

, (x ≥ 0). If α is a positive real number such that a = (fog)'(α) and b = (fog)(α), then

(A) $a\alpha^{2} + b\alpha - a = 2\alpha^{2}$
(B) $a\alpha^{2} - b\alpha - a = 0$
(C) $a\alpha^{2} - b\alpha - a = 1$
(D) $a\alpha^{2} + b\alpha + a = 0$

Q86·MathematicsSingle correctJEE Main 2019

Let f: R → R be differentiable at c ∈ R and f(c) = 0. If g(x) = |f(x)|, then at x = c, g is:

(A)differentiable if f′(c) = 0
(B)differentiable if f′(c) ≠ 0
(C)not differentiable
(D)not differentiable if f′(c) = 0

Q87·MathematicsSingle correctJEE Main 2019

Let

f(x)=15-|x-10|; x\in R

. then the set of all values of x, at which the function,

g(x)=f(f(x))

is not differentiable, is:

(A) $\{5,10,15\}$
(B) $\{10\}$
(C) $\{5,10,15,20\}$
(D) $\{10,15\}$

Q88·MathematicsSingle correctJEE Main 2019

If

f(1)=1, f'(1)=3

, then the derivative of

f(f(f(x)))+(f(x))^{2}

at

x=1

is:

(A)33
(B)15
(C)9
(D)12

Q89·MathematicsSingle correctJEE Main 2019

Let f be a differentiable function such that f(1) = 2 and f'(x) = f(x) for all x∈R. If h(x) = f(f(x)), then h'(1) is equal to :

(A) $2e^{2}$
(B) $4e$
(C) $2e$
(D) $4e^{2}$

Q90·MathematicsSingle correctJEE Main 2019

For x > 1, if

(2x)^{2y}=4e^{2x-2y}

, then

(1+\log_{e}2x)^{2}\frac{dy}{dx}

is equal to:

(A) $\frac{x\log_{e}2x-\log_{e}2}{x}$
(B) $\log_{e}2x$
(C) $\frac{x\log_{e}2x+\log_{e}2}{x}$
(D) $x\log_{e}2x$

Q91·MathematicsSingle correctJEE Main 2019

Let S be the set of all points in

(-\pi, \pi)

at which the function,

f(x)=\min\{\sin x, \cos x\}

is non-differentiable. Then S is a subset of which of the following?

(A) $\left\{-\frac{\pi}{4},0,\frac{\pi}{4}\right\}$
(B) $\left\{-\frac{3\pi}{4},-\frac{\pi}{4},\frac{3\pi}{4},\frac{\pi}{4}\right\}$
(C) $\left\{-\frac{\pi}{2},-\frac{\pi}{4},\frac{\pi}{4},\frac{\pi}{2}\right\}$
(D) $\left\{-\frac{3\pi}{4},-\frac{\pi}{2},\frac{\pi}{2},\frac{3\pi}{4}\right\}$

Q92·MathematicsSingle correctJEE Main 2019

If

x\log_{e}(\log_{e}x)-x^{2}+y^{2}=4\,(y>0)

, then

\frac{dy}{dx}

at x = e is equal to:

(A) $\frac{(1+2e)}{2\sqrt{4+e^{2}}}$
(B) $\frac{(2e-1)}{2\sqrt{4+e^{2}}}$
(C) $\frac{(1+2e)}{\sqrt{4+e^{2}}}$
(D) $\frac{e}{\sqrt{4+e^{2}}}$

Q93·MathematicsSingle correctJEE Main 2019

Let K be the set of all real values of x where the function

f(x)=\sin|x|-|x|+2(x-\pi)\cos|x|

is not differentiable. Then the set K is equal to:

(A) $\phi$ (en empty set)
(B) $\{\pi\}$
(C) $\{0\}$
(D) $\{0,\pi\}$

Q94·MathematicsSingle correctJEE Main 2019

Let

f(x)=\begin{cases}-1, & -2\le x<0\\ x^{2}-1, & 0\le x\le 2\end{cases}

and

g(x)=|f(x)|+f(|x|)

, Then, in the interval

(-2,2)

, g is:

(A)differentiable at all points
(B)not continuous
(C)not differentiable at two points
(D)not differentiable at one point

Q95·MathematicsSingle correctJEE Main 2019

Let

f : R \to R

be a function such that

f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3)

,

x \in R

. Then

f(2)

equals:

(A)-4
(B)30
(C)-2
(D)8

Q96·MathematicsSingle correctJEE Main 2019

Let

f(x) = \begin{cases} \max\{|x|, x^2\}, & |x| \leq 2 \\ 8 - 2|x|, & 2 < |x| \leq 4 \end{cases}

. Let S be the set of points in the interval (−4, 4) at which f is not differentiable. Then S:

(A)is an empty set
(B)equals $\{-2, -1, 0, 1, 2\}$
(C)equals $\{-2, -1, 1, 2\}$
(D)equals $\{-2, 2\}$

Q97·MathematicsSingle correctJEE Main 2019

If

x = 3\tan t

and

y = 3\sec t

, then the value of

\frac{d^{2}y}{dx^{2}}

at

t = \frac{\pi}{4}

, is:

(A) $\frac{3}{2\sqrt{2}}$
(B) $\frac{1}{3\sqrt{2}}$
(C) $\frac{1}{6}$
(D) $\frac{1}{6\sqrt{2}}$

Q98·MathematicsSingle correctJEE Main 2019

Let f be a differentiable function R to R such that

\left| f(x) - f(y) \right| \le 2\left| x - y \right|^{\frac{3}{2}}

, for all

x, y \in R

. If

f(0) = 1

then

\int_{0}^{1} f^{2}(x) dx

is equal to

(A) $0$
(B) $\frac{1}{2}$
(C) $2$
(D) $1$

Q99·MathematicsSingle correctJEE Advanced 2018

Let

f_{1} : R \to R

,

f_{2} : \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \to R

,

f_{3} : (-1, e^{\pi/2} - 2) \to R

and

f_{4} : R \to R

be functions defined by (i)

f_{1}(x) = \sin\left(\sqrt{1 - e^{-x^{2}}}\right)

, (ii)

f_{2}(x) = \begin{cases} \frac{|\sin x|}{\tan^{-1} x} & \text{if } x \neq 0 \\\\ 1 & \text{if } x = 0 \end{cases}

, where the inverse trigonometric function

\tan^{-1}x

assumes values in

\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

, (iii)

f_{3}(x) = [\sin(\log_{e}(x + 2))]

, where, for

t \in R

,

[t]

denotes the greatest integer less than or equal to

t

, (iv)

f_{4}(x) = \begin{cases} x^{2}\sin\left(\frac{1}{x}\right) & \text{if } x \neq 0 \\\\ 0 & \text{if } x = 0 \end{cases}

. The correct option is :

LIST-I	LIST-II
P.The function $f_{1}$ is	1.NOT continuous at $x = 0$
Q.The function $f_{2}$ is	2.continuous at $x = 0$ and NOT differentiable at $x = 0$
R.The function $f_{3}$ is	3.differentiable at $x = 0$ and its derivative is NOT continuous at $x = 0$
S.The function $f_{4}$ is	4.differentiable at $x = 0$ and its derivative is continuous at $x = 0$

(A)P $\to$ 2; Q $\to$ 3; R $\to$ 1; S $\to$ 4
(B)P $\to$ 4; Q $\to$ 1; R $\to$ 2; S $\to$ 3
(C)P $\to$ 4; Q $\to$ 2; R $\to$ 1; S $\to$ 3
(D)P $\to$ 2; Q $\to$ 1; R $\to$ 4; S $\to$ 3

Q100·MathematicsMultiple correctJEE Advanced 2016

Let

a, b \in \mathbb{R}

and

f : \mathbb{R} \to \mathbb{R}

be defined by

f(x) = a \cos(|x^{3} - x|) + b|x| \sin(|x^{3} + x|)

. Then f is

(A)differentiable at $x = 0$ if $a = 0$ and $b = 1$
(B)differentiable at $x = 1$ if $a = 1$ and $b = 0$
(C)NOT differentiable at $x = 0$ if $a = 1$ and $b = 0$
(D)NOT differentiable at $x = 1$ if $a = 1$ and $b = 1$

Q101·MathematicsMultiple correctJEE Advanced 2016

Let

f : \left[ -\frac{1}{2}, 2 \right] \to \mathbb{R}

and

g : \left[ -\frac{1}{2}, 2 \right] \to \mathbb{R}

be functions defined by

f(x) = [x^{2} - 3]

and

g(x) = |x| f(x) + |4x - 7| f(x)

, where

[y]

denotes the greatest integer less than or equal to y for

y \in \mathbb{R}

. Then

(A)f is discontinuous exactly at three points in $\left[ -\frac{1}{2}, 2 \right]$
(B)f is discontinuous exactly at four points in $\left[ -\frac{1}{2}, 2 \right]$
(C)g is NOT differentiable exactly at four points in $\left( -\frac{1}{2}, 2 \right)$
(D)g is NOT differentiable exactly at five points in $\left( -\frac{1}{2}, 2 \right)$

Q102·MathematicsMultiple correctJEE Advanced 2016

Let

f : \mathbb{R} \to \mathbb{R}

,

g : \mathbb{R} \to \mathbb{R}

and

h : \mathbb{R} \to \mathbb{R}

be differentiable functions such that

f(x) = x^{3} + 3x + 2

,

g(f(x)) = x

and

h(g(g(x))) = x

for all

x \in \mathbb{R}

. Then

(A) $g'(2) = \frac{1}{15}$
(B) $h'(1) = 666$
(C) $h(0) = 16$
(D) $h(g(3)) = 36$

Q103·MathematicsMultiple correctJEE Advanced 2015

Let

g : \mathbb{R} \to \mathbb{R}

be a differential function with

g(0) = 0

,

g'(0) = 0

and

g'(1) \ne 0

. Let

f(x) = \begin{cases} \dfrac{x}{|x|}g(x), & x \ne 0 \\ 0, & x = 0 \end{cases}

and

h(x) = e^{|x|}

for all

x \in \mathbb{R}

. Let

(f \circ h)(x)

denote

f(h(x))

and

(h \circ f)(x)

denote

h(f(x))

. Then which of the following is (are) true?

(A) $f$ is differentiable at $x = 0$
(B) $h$ is differentiable at $x = 0$
(C) $f \circ h$ is differentiable at $x = 0$
(D) $h \circ f$ is differentiable at $x = 0$

Q104·MathematicsIntegerJEE Advanced 2014

Let

f: \mathbb{R} \to \mathbb{R}

and

g: \mathbb{R} \to \mathbb{R}

be respectively given by

f(x) = |x| + 1

and

g(x) = x^2 + 1

. Define

h: \mathbb{R} \to \mathbb{R}

by

h(x) = \begin{cases} \max \{f(x), g(x)\} & \text{if } x \le 0 \\ \min \{f(x), g(x)\} & \text{if } x > 0 \end{cases}

. Then number of points at which

h(x)

is not differentiable is __________

Q105·MathematicsMultiple correctJEE Advanced 2014

Let

f: [a, b] \to [1, \infty)

be a continuous function and let

g: \mathbb{R} \to \mathbb{R}

be defined as

g(x) = \begin{cases} 0 & \text{if } x < a, \\ \int_{a}^{x} f(t) dt & \text{if } a \le x \le b, \\ \int_{a}^{b} f(t) dt & \text{if } x > b \end{cases}

Then

(A) $g(x)$ is continuous but not differentiable at $a$
(B) $g(x)$ is differentiable on $\mathbb{R}$
(C) $g(x)$ is continuous but not differentiable at $b$
(D) $g(x)$ is continuous and differentiable at either $a$ or $b$ but not both

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