EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 19th April 2014>Q 68

HARD

JEE Main

IMPORTANT

Earn 100

Let $f : R \to R$ be a function such that $|f (x)| \leq x^{2},$ for all $x \in R .$ Then, at $x = 0,$ $f$ is

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Important Questions on Continuity and Differentiability

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 9th April 2014>Q 69

If the Rolle's theorem holds for the function

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

in the interval

[- 1, 1]

for the point

c = \frac{1}{2}

, then the value of

2 a + b

is:

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 6th April 2014>Q 70

If

f & g

are differentiable functions in

[0, 1]

satisfying

f (0) = 2 = g (1), g (0) = 0 & f (1) = 6,

then for some

c \in]0, 1[

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 24 June 2022 Shift 1>Q 71

The number of points where the function

f (x) = \{\begin{cases} |2 x^{2} - 3 x - 7| & if x ⩽ - 1 \\ [4 x^{2} - 1] & if - 1 < x < 1 \\ |x + 1| + |x - 2| & if x ⩾ 1 \end{cases}

, where

[t]

denotes the greatest integer

⩽ t

, is discontinuous is ______

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 24 June 2022 Shift 2>Q 72

Let

f (x) = \{\begin{cases} \frac{\sin (x - [x])}{x - [x]}, & x \in (- 2, - 1) \\ \max (2 x, 3 [|x|]), & |x| < 1 \\ 1, & otherwise \end{cases}

where

[t]

denotes greatest integer

\leq t

. If

m

is the number of points where

f

is continuous and

n

is the number of points where

f

is not differentiable, the ordered pair

(m, n)

is:

MEDIUM

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 25 June 2022 Shift 2>Q 73

Let

f (x) = [2 x^{2} + 1]

and

g (x) = \{\begin{cases} 2 x - 3, & x < 0 \\ 2 x + 3, & x \geq 0 \end{cases}

, where

[t]

is the greatest integer

\leq t

. Then, in the open interval

(- 1, 1)

, the number of points where

f o g

is discontinuous is equal to ______.

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 26 June 2022 Shift 2>Q 74

Let

f (x) = \min \{1, 1 + x \sin x\}, 0 \leq x \leq 2 π

. If

m

is the number of points, where

f

is not differentiable and

n

is the number of points, where

f

is not continuous, then the ordered pair

(m, n)

is equal to

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 26 June 2022 Shift 1>Q 75

f, g : R \to R

be two real valued function defined as

f (x) = \{\begin{matrix} - |x + 3| & , & x < 0 \\ e^{x} & , & x \geq 0 \end{matrix}

and

g (x) = \{\begin{cases} x^{2} + k_{1} x & , & x < 0 \\ 4 x + k_{2} & , & x \geq 0 \end{cases}

, where

k_{1}

and

k_{2}

are real constants. If

g o f

is differentiable at

x = 0

, then

g o f (- 4) +

g o f (4)

is equal to

HARD

JEE Main

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 25 - Continuity and Differentiability>JEE Main - 28 June 2022 Shift 1>Q 76

Let

f : ℝ \to ℝ

be defined as

f (x) = [\begin{array}{l} [e^{x}], & x < 0 \\ a e^{x} + [x - 1], & 0 \leq x < 1 \\ b + [\sin (π x)], & 1 \leq x < 2 \\ [e^{- x}] - c, & x \geq 2 \end{array}

where

a, b, c \in ℝ

and

[t]

denotes greatest integer less than or equal to

t

. Then, which of the following statements is true?