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

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

HARD

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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 $[0, \infty)$

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Important Questions on Continuity and Differentiability

MEDIUM

Mathematics>Calculus>Continuity and Differentiability>Properties of Differentiable Function

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>Calculus>Continuity and Differentiability>Properties of Differentiable Function

What is the derivative of

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

with respect to

x ?

MEDIUM

Mathematics>Calculus>Continuity and Differentiability>Properties of Differentiable Function

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 :

HARD

Mathematics>Calculus>Continuity and Differentiability>Properties of Differentiable Function

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 ..... .

MEDIUM

Mathematics>Calculus>Continuity and Differentiability>Properties of Differentiable Function

The function

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

is not differentiable at

MEDIUM

Mathematics>Calculus>Continuity and Differentiability>Properties of Differentiable Function

Which function is differentiable and why?

HARD

Mathematics>Calculus>Continuity and Differentiability>Properties of Differentiable Function

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)

MEDIUM

Mathematics>Calculus>Continuity and Differentiability>Properties of Differentiable Function

[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 ______.

MEDIUM

Mathematics>Calculus>Continuity and Differentiability>Properties of Differentiable Function

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>Calculus>Continuity and Differentiability>Properties of Differentiable Function

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:

HARD

Mathematics>Calculus>Continuity and Differentiability>Properties of Differentiable Function

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>Calculus>Continuity and Differentiability>Properties of Differentiable Function

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)}

EASY

Mathematics>Calculus>Continuity and Differentiability>Properties of Differentiable Function

Every differentiable function is_____.

EASY

Mathematics>Calculus>Continuity and Differentiability>Properties of Differentiable Function

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>Calculus>Continuity and Differentiability>Properties of Differentiable Function

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

MEDIUM

Mathematics>Calculus>Continuity and Differentiability>Properties of Differentiable Function

Let

f (x) = \{\begin{matrix} m a x (|x|, x^{2}), & |x| \leq 2 \\ 8 - 2 |x|, & 2 < |x| \leq 4 \end{matrix} .

Let

S

be the set of points in the interval

(- 4, 4)

at which

f

is not differentiable. Then

S

HARD

Mathematics>Calculus>Continuity and Differentiability>Properties of Differentiable Function

If the function

g (x) = {\begin{matrix} k \sqrt{x + 1}, & 0 \leq x \leq 3 \\ m x + 2, & 3 < x \leq 5 \end{matrix}

is differentiable, then the value of

k + m

HARD

Mathematics>Calculus>Continuity and Differentiability>Properties of Differentiable Function

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

HARD

Mathematics>Calculus>Continuity and Differentiability>Properties of Differentiable Function

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

HARD

Mathematics>Calculus>Continuity and Differentiability>Properties of Differentiable Function

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:

Let fx be a non-negative differentiable function on [0,∞) such that f0=0 and f'x≤2fx for all x>0. Then, on [0,∞)

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

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 $[0, \infty)$