Differentiability of a Function

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JEE Main 2008

LEVELJEE Main

Let $f (x) = {(x - 1) sin \frac{1}{x - 1}, 0, if x \neq = 1 if x = 1$ . Then which one of the following is true?

(A)

f is neither differentiable at x = 0 nor at x = 1

(B)

f is differentiable at x = 0 and at x = 1

(C)

f is differentiable at x = 0 but not at x = 1

(D)

f is differentiable at x = 1 but not at x = 0

JEE Main 2007

LEVELJEE Main

Let $f : R \to R$ be a function defined by $f (x) = min {x + 1, ∣ x ∣ + 1}$ . Then which of the following is true?

(A)

f(x) is differentiable everywhere

(B)

f(x) is not differentiable at x = 0

(C)

f(x) \geq 1 for all x \in R

(D)

f(x) is not differentiable at x = 1

JEE Main 2006

LEVELJEE Main

The set of points where $f (x) = \frac{x}{1 + ∣ x ∣}$ is differentiable is

(A)

(- \infty, 0) \cup (0, \infty)

(B)

(- \infty, - 1) \cup (- 1, \infty)

(C)

(- \infty, \infty)

(D)

(0, \infty)

JEE Advanced 2015

LEVELJEE Advanced

Let $g : R \to R$ be a differentiable function with $g (0) = 0, g^{'} (0) = 0$ and $g^{'} (1) \neq = 0$ . Let $f (x) = {\frac{x}{∣ x ∣} g (x), 0, x \neq = 0 x = 0$ and $h (x) = e^{∣ x ∣}$ for all $x \in R$ . Let $(f o h) (x)$ denote $f (h (x))$ and $(h o f) (x)$ denote $h (f (x))$ . Then which of the following is (are) true?

* Multiple Correct Options

(A)

f

is differentiable at

x = 0

(B)

h

is differentiable at

x = 0

(C)

f o h

is differentiable at

x = 0

(D)

h o f

is differentiable at

x = 0

JEE Advanced 2012

LEVELJEE Main

Let $f (x) = {x^{2} cos \frac{π}{x} 0 x \neq = 0 x = 0, x \in R$ then $f$ is

(A)

differentiable both at

x = 0

and at

x = 2

(B)

differentiable at

x = 0

but not differentiable at

x = 2

(C)

not differentiable at

x = 0

but differentiable at

x = 2

(D)

differentiable neither at

x = 0

nor at

x = 2

JEE Advanced 2001

LEVELJEE Main

The left-hand derivative of $f (x) = [x] sin (π x)$ at $x = k, k$ an integer, is

(A)

(- 1)^{k} (k - 1) π

(B)

(- 1)^{k - 1} (k - 1) π

(C)

(- 1)^{k} k π

(D)

(- 1)^{k - 1} k π

JEE Advanced 2001

LEVELJEE Main

Let $f : R \to R$ be a function defined by $f (x) = max {x, x^{3}}$ . The set of all points where $f (x)$ is NOT differentiable is

(A)

{- 1, 1}

(B)

{- 1, 0}

(C)

{0, 1}

(D)

{- 1, 0, 1}

JEE Advanced 2001

LEVELJEE Main

Which of the following functions is differentiable at $x = 0$ ?

(A)

cos (∣ x ∣) + ∣ x ∣

(B)

cos (∣ x ∣) - ∣ x ∣

(C)

sin (∣ x ∣) + ∣ x ∣

(D)

sin (∣ x ∣) - ∣ x ∣

JEE Advanced 1999

LEVELJEE Main

The function $f (x) = (x^{2} - 1) ∣ x^{2} - 3 x + 2∣ + cos (∣ x ∣)$ is NOT differentiable at

(A)

- 1

Let $h (x) = min {x, x^{2}}$ , for every real number of $x$ , Then

* Multiple Correct Options

(A)

h

is continuous for all

x

(B)

h

is differentiable for all

x

(C)

h^{'} (x) = 1

, for all

x > 1

(D)

h

is not differentiable at two values of

x

JEE Advanced 1985

LEVELJEE Main

If $f (x) = x (x - x + 1)$ , then

(A)

f (x)

is continuous but not differentiable at

x = 0

(B)

f (x)

is differentiable at

x = 0

(C)

f (x)

is not differentiable at

x = 0

If $f (x - y) = f (x) \cdot g (y) - f (y) \cdot g (x)$ and $g (x - y) = g (x) \cdot g (y) + f (x) \cdot f (y)$ for all $x, y \in R$ . If right hand derivative at $x = 0$ exists for $f (x)$ . Find derivative of $g (x)$ at $x = 0$ .

JEE Advanced 1987

LEVELJEE Main

Let $f (x)$ be a function satisfying the condition $f (- x) = f (x)$ for all real $x$ . If $f^{'} (0)$ exists, find its value.

JEE Advanced 1983

LEVELJEE Main

For the function $f (x) = {\frac{x}{1 + e ^{1/ x}}, 0, x \neq = 0 x = 0$ , the derivative from the right, $f^{'} (0^{+}) = .........$ , and the derivative from the left, $f^{'} (0^{-}) = .........$

JEE Advanced 1979

LEVELJEE Main

Find the derivative of $f (x) = {\frac{x - 1}{2 x ^{2} - 7 x + 5} - \frac{1}{3} when x \neq = 1 when x = 1$ at $x = 1$ .

JEE Advanced 2001

LEVELJEE Main

Let $α \in R$ . Prove that a function $f : R \to R$ is differentiable at $α$ if and only if there is a function $g : R \to R$ which is continuous at $α$ and satisfies $f (x) - f (α) = g (x) (x - α)$ for all $x \in R$ .

JEE Main 2018 (Paper 1)

LEVELJEE Advanced

Let $S = {t \in R : f (x) = ∣ x - π ∣ \cdot (e^{∣ x ∣} - 1) sin ∣ x ∣ is not differentiable at t}$ . Then the set $S$ is equal to :

(A)

{0, \pi}

(B)

ϕ (an empty set)

JEE Main 2018 (15 April Shift 1)

LEVELJEE Main

Let $S = {(λ, μ) \in R \times R : f (t) = (∣ λ ∣ e^{∣ t ∣} - μ) sin (2∣ t ∣), t \in R, is a differentiable function}$ . Then S is a subset of :

(A)

[0, \infty) \times R

(B)

R \times (- \infty, 0)

(C)

R \times [0, \infty)

(D)

(- \infty, 0) \times R

JEE Main 2019 (10 January Shift 1)

LEVELJEE Main

Let $f (x) = {max {∣ x ∣, x^{2}}, 8 - 2∣ x ∣, ∣ x ∣ \leq 2 2 < ∣ x ∣ \leq 4$ . 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, 1, 2\}

(C)

equals \{-2, -1, 0, 1, 2\}

(D)

equals \{-2, 2\}

JEE Main 2019 (9 January)

LEVELJEE Main

Let $f$ be a differentiable function from $R$ to $R$ such that $∣ f (x) - f (y) ∣ \leq 2∣ x - y ∣^{3/2}$ , for all $x, y \in R$ . If $f (0) = 1$ then $\int_{0}^{1} f^{2} (x) d x$ is equal to

JEE Main 2019 (12 January Shift 1)

LEVELJEE Main

Let $S$ be the set of all points in $(- π, π)$ at which the function, $f (x) = min {sin x, cos x}$ is not differentiable. Then $S$ is a subset of which of the following?

(A)

{- \frac{3 π}{4}, - \frac{π}{4}, \frac{π}{4}, \frac{3 π}{4}}

(B)

{- \frac{3 π}{4}, - \frac{π}{2}, \frac{π}{2}, \frac{3 π}{4}}

(C)

{- \frac{π}{2}, - \frac{π}{4}, \frac{π}{4}, \frac{π}{2}}

(D)

{- \frac{π}{4}, 0, \frac{π}{4}}

JEE Main 2019 (10 January)

LEVELJEE Main

Let $f : (- 1, 1) \to R$ be a function defined by $f (x) = max {- ∣ x ∣, - 1 - x^{2}}$ . If K be the set of all points at which f is not differentiable, then K has exactly :

JEE Main 2019 (09 April Shift 1)

LEVELJEE Main

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, 20}

(B)

{10, 15}

(C)

{5, 10, 15}

(D)

{10}

JEE Main 2019 (10 April Shift 1)

LEVELJEE Main

Let $f : R \to R$ be differentiable at $c \in R$ and $f (c) = 0$ . If $g (x) = ∣ f (x) ∣$ , then at $x = c$ , $g$ is :

(A)

differentiable if

f^{'} (c) = 0

f^{'} (c) \neq = 0

(D)

not differentiable if

f^{'} (c) = 0

JEE Main 11 Jan 2019 (Evening)

LEVELJEE Main

Let $K$ be the set of all real values of $x$ where the function $f (x) = sin ∣ x ∣ - ∣ x ∣ + 2 (x - π) cos ∣ x ∣$ is not differentiable. Then the set $K$ is equal to:

(A)

ϕ

(an empty set)

(B)

{π}

(C)

{0}

(D)

{0, π}

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)={(x−1)sinx−11​,0,​if x=1if x=1​. Then which one of the following is true?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be a function defined by f(x)=min{x+1,∣x∣+1}. Then which of the following is true?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The set of points where f(x)=1+∣x∣x​ is differentiable is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)={x2cosxπ​0​x=0x=0​,x∈R then f is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The left-hand derivative of f(x)=[x]sin(πx) at x=k,k an integer, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be a function defined by f(x)=max{x,x3}. The set of all points where f(x) is NOT differentiable is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Which of the following functions is differentiable at x=0?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The function f(x)=(x2−1)∣x2−3x+2∣+cos(∣x∣) is NOT differentiable at

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let h(x)=min{x,x2}, for every real number of x, Then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x)=x(x​−x+1​), then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If f(x−y)=f(x)⋅g(y)−f(y)⋅g(x) and g(x−y)=g(x)⋅g(y)+f(x)⋅f(y) for all x,y∈R. If right hand derivative at x=0 exists for f(x). Find derivative of g(x) at x=0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x) be a function satisfying the condition f(−x)=f(x) for all real x. If f′(0) exists, find its value.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For the function f(x)={1+e1/xx​,0,​x=0x=0​, the derivative from the right, f′(0+)=........., and the derivative from the left, f′(0−)=.........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the derivative of f(x)={2x2−7x+5x−1​−31​​when x=1when x=1​ at x=1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S={t∈R:f(x)=∣x−π∣⋅(e∣x∣−1)sin∣x∣ is not differentiable at t}. Then the set S is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S={(λ,μ)∈R×R:f(t)=(∣λ∣e∣t∣−μ)sin(2∣t∣),t∈R, is a differentiable function}. Then S is a subset of :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)={max{∣x∣,x2},8−2∣x∣,​∣x∣≤22<∣x∣≤4​. Let S be the set of points in the interval (-4, 4) at which f is not differentiable. Then S:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f be a differentiable function from R to R such that ∣f(x)−f(y)∣≤2∣x−y∣3/2, for all x,y∈R. If f(0)=1 then ∫01​f2(x)dx is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S be the set of all points in (−π,π) at which the function, f(x)=min{sinx,cosx} is not differentiable. Then S is a subset of which of the following?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:(−1,1)→R be a function defined by f(x)=max{−∣x∣,−1−x2​}. If K be the set of all points at which f is not differentiable, then K has exactly :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f(x)=15−∣x−10∣;x∈R. Then the set of all values of x, at which the function, g(x)=f(f(x)) is not differentiable, is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let f:R→R be differentiable at c∈R and f(c)=0. If g(x)=∣f(x)∣, then at x=c, g is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let K be the set of all real values of x where the function f(x)=sin∣x∣−∣x∣+2(x−π)cos∣x∣ is not differentiable. Then the set K is equal to:

Let $f (x) = {(x - 1) sin \frac{1}{x - 1}, 0, if x \neq = 1 if x = 1$ . Then which one of the following is true?

Let $f : R \to R$ be a function defined by $f (x) = min {x + 1, ∣ x ∣ + 1}$ . Then which of the following is true?

The set of points where $f (x) = \frac{x}{1 + ∣ x ∣}$ is differentiable is

Let $f (x) = {x^{2} cos \frac{π}{x} 0 x \neq = 0 x = 0, x \in R$ then $f$ is

The left-hand derivative of $f (x) = [x] sin (π x)$ at $x = k, k$ an integer, is

Let $f : R \to R$ be a function defined by $f (x) = max {x, x^{3}}$ . The set of all points where $f (x)$ is NOT differentiable is

Which of the following functions is differentiable at $x = 0$ ?

The function $f (x) = (x^{2} - 1) ∣ x^{2} - 3 x + 2∣ + cos (∣ x ∣)$ is NOT differentiable at

Let $h (x) = min {x, x^{2}}$ , for every real number of $x$ , Then

If $f (x) = x (x - x + 1)$ , then

If $f (x - y) = f (x) \cdot g (y) - f (y) \cdot g (x)$ and $g (x - y) = g (x) \cdot g (y) + f (x) \cdot f (y)$ for all $x, y \in R$ . If right hand derivative at $x = 0$ exists for $f (x)$ . Find derivative of $g (x)$ at $x = 0$ .

Let $f (x)$ be a function satisfying the condition $f (- x) = f (x)$ for all real $x$ . If $f^{'} (0)$ exists, find its value.

For the function $f (x) = {\frac{x}{1 + e ^{1/ x}}, 0, x \neq = 0 x = 0$ , the derivative from the right, $f^{'} (0^{+}) = .........$ , and the derivative from the left, $f^{'} (0^{-}) = .........$

Find the derivative of $f (x) = {\frac{x - 1}{2 x ^{2} - 7 x + 5} - \frac{1}{3} when x \neq = 1 when x = 1$ at $x = 1$ .

Let $S = {t \in R : f (x) = ∣ x - π ∣ \cdot (e^{∣ x ∣} - 1) sin ∣ x ∣ is not differentiable at t}$ . Then the set $S$ is equal to :

Let $S = {(λ, μ) \in R \times R : f (t) = (∣ λ ∣ e^{∣ t ∣} - μ) sin (2∣ t ∣), t \in R, is a differentiable function}$ . Then S is a subset of :

Let $f (x) = {max {∣ x ∣, x^{2}}, 8 - 2∣ x ∣, ∣ x ∣ \leq 2 2 < ∣ x ∣ \leq 4$ . Let S be the set of points in the interval (-4, 4) at which f is not differentiable. Then S:

Let $f$ be a differentiable function from $R$ to $R$ such that $∣ f (x) - f (y) ∣ \leq 2∣ x - y ∣^{3/2}$ , for all $x, y \in R$ . If $f (0) = 1$ then $\int_{0}^{1} f^{2} (x) d x$ is equal to

Let $S$ be the set of all points in $(- π, π)$ at which the function, $f (x) = min {sin x, cos x}$ is not differentiable. Then $S$ is a subset of which of the following?

Let $f : (- 1, 1) \to R$ be a function defined by $f (x) = max {- ∣ x ∣, - 1 - x^{2}}$ . If K be the set of all points at which f is not differentiable, then K has exactly :

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 :

Let $f : R \to R$ be differentiable at $c \in R$ and $f (c) = 0$ . If $g (x) = ∣ f (x) ∣$ , then at $x = c$ , $g$ is :

Let $K$ be the set of all real values of $x$ where the function $f (x) = sin ∣ x ∣ - ∣ x ∣ + 2 (x - π) cos ∣ x ∣$ is not differentiable. Then the set $K$ is equal to: