Let f x be a non-negative differentiable function on [ 0 , ∞ ) such that f 0 = 0 and f ' x ≤ 2 f x for all x > 0 . Then, on [ 0 , ∞ )

f x is always a constant function

For each x ∈ R , let f ( x ) = | x - 1 | , g ( x ) = cos x and ϕ ( x ) = f ( g ( 2 sin x ) ) - g ( f ( x ) . Then ϕ is

differentiable at each point of R

differentiable only in - π 2 , π 2

If f x + y = f x + f y + x y + x 2 y 2 , ∀ x , y ∈ R and f ' 0 = 0 , then

f is differentiable for all x ∈ R .

f need not be differentiable at every non-zero x .

f is twice differentiable at x = 0 .

HARD

Earn 100

Let $f, g : ℝ \to ℝ$ be such that $f (x) = x^{k} g (x)$ where $k$ is a positive integer, Suppose $g$ is continuous at $x = 0$ . Then

50% studentsanswered this correctly

Important Questions on Continuity and Differentiability

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

Let

f (x)

be a non-negative differentiable function on

[0, \infty)

such that

f (0) = 0

and

f^{'} (x) \leq 2 f (x)

for all

x > 0 .

Then, on

[0, \infty)

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

Let

f

be a differentiable function from

R

R

such that

|f (x) - f (y)| \leq 2 {|x - y|}^{3 / 2},

for all

x, y \in R .

f (0) = 1

then

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

is equal to

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

f (x) = \{\begin{array}{cll} \frac{1}{| x |} & ; & | x | \geq 1 \\ a x^{2} + b & ; & | x | < 1 \end{array}

is differentiable at every point of the domain, then the values of

a

and

b

are respectively:

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

Let

f : N \to N

be a function such that

f (m + n) = f (m) + f (n)

for every

m, n \in N .

f (6) = 18

then

f (2) \cdot f (3)

is equal to :

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

What is the derivative of

\log_{10} (5 x^{2} + 3)

with respect to

x ?

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

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

for all

x, y, z

and

f (2) = 4, f^{'} (0) = 3,

then the value of

f^{'} (2)

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

The number of points, at which the function

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

is not differentiable, is

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

[t]

denotes the greatest integer

\leq t

, then number of points, at which the function

f (x) = 4 |2 x + 3| +

9 [x + \frac{1}{2}] - 12 [x + 20]

is not differentiable in the open interval

(- 20, 20)

, is ______.

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

Let $f : (- 1, 1) \to R$ be a differentiable function satisfying ${(f^{'} (x))}^{4} = 16 {(f (x))}^{2}$ for all $x \in (- 1, 1)$ $f (0) = 0$ . The number of such functions is

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

The function

(x^{2} - 9) |x^{2} - 7 x + 12| + \cos (| x |)

is not differentiable at

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

Consider the following statements.
a) If a function is differentiable at a point

' p'

then it is not continuous at

' p'

b) If a function is not continuous at

x = a,

then it is not differentiable at

x = a

c) If

f (x) = |x|

then

f (x)

is not differentiable but continuous on

ℝ

d) If

f (x) = x - [x],

then

f' (1) = 1

Which of the above statements are (is) correct?

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

Suppose that a function

f : R \to R

satisfies

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

for all

x, y ε R

and

f (1) = 3

. If

\sum_{i = 1}^{n} f (i) = 363

, then

n

is equal to ..... .

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

Every differentiable function is_____.

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

For each

x \in R,

let

f (x) = | x - 1 |, g (x) = \cos x

and

ϕ (x) = f (g (2 \sin x)) - g (f (x) .

Then

ϕ

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

Which function is differentiable and why?

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

Let

f

be any function defined on

R

and let it satisfy the condition:

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

. If

f (0) = 1,

then :

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

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

and

Σ_{x = 1}^{\infty} f (x) = 2, x, y \in N

, where

N

is the set of all natural numbers, then the value of

\frac{f (4)}{f (2)}

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

If the function

f (x) = \{\begin{array}{cl} k_{1} (x - π)^{2} - 1, & x \leq π \\ k_{2} \cos x, & x > π \end{array}

is twice differentiable, then the ordered pair

(k_{1}, k_{2})

is equal to:

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

f (x + y) = f (x) + f (y) + |x| y + x^{2} y^{2}, \forall x, y \in R

and

f^{'} (0) = 0

, then

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Properties of Differentiable Functions

Represent the union of two sets by Venn diagram for each of the following.

$X = {x | x$ is a prime number between $80$ and $100}$

$Y = {y | y$ is an odd number between $90$ and $100}$

Let f,g : ℝ→ℝ be such that fx=xkgx where k is a positive integer, Suppose g is continuous at x=0. Then

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

Let $f, g : ℝ \to ℝ$ be such that $f (x) = x^{k} g (x)$ where $k$ is a positive integer, Suppose $g$ is continuous at $x = 0$ . Then