For x ∈ R , f x = log ⁡ 2 - sin ⁡ x and g x = f f x , then

g is differentiable at x = 0 and g ' 0 = - sin ⁡ log ⁡ 2

g is not differentiable at x = 0

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

not differentiable if f ' c = 0

If f x = x for x ≤ 0 0 for x > 0 then f ( x ) at x = 0 is

Continuous but not differentiable

Not continuous but differentiable

Not continuous and not differentiable

The function f x = π 4 + tan - 1 x , x ≤ 1 1 2 x - 1 , x > 1 is :

continuous on R - 1 and differentiable on R - - 1 , 1 .

both continuous and differentiable on R - 1

continuous on R - - 1 and differentiable on R - - 1 , 1

both continuous and differentiable on R - - 1

The function f x = e x is

continuous everywhere but not differentiable at x = 0

continuous and differentiable everywhere

If f x = x e - 1 x + 1 x ; i f x ≠ 0 0 ; i f x = 0 then which of the following is correct?

f x is continuous and f ' 0 does not exist

f x is continuous and f ′ 0 also exists

f x = x sin 1 x , x ≠ 0 0 , x = 0 at x = 0 is

continuous but not differentiable

continuous as well as differentiable

differentiable but not continuous

neither continuous nor differentiable

HARD

Earn 100

Let $f : R \to R$ be defined as

$f (x) = \{\begin{matrix} \frac{a - b \cos 2 x}{x^{2}}; x < 0 \\ x^{2} + c x + 2; 0 \leq x \leq 1 \\ 2 x + 1; x > 1 \end{matrix}$

If $f$ is continuous everywhere in $R$ and $m$ is the number of points where $f$ is NOT differential then $m + a + b + c$ equals:

34.73% studentsanswered this correctly

Important Questions on Continuity and Differentiability

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

For

x \in R, f (x) = |\log 2 - \sin x|

and

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

, then

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

Directions: Each of the following sentences in this section has a blank space and four words or group of words given after the sentence. Select the word or group of words you consider most appropriate for the blank space and indicate your response on the Answer Sheet accordingly.

Every rash driver becomes a ____ killer.

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

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:

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

f (x) = [x]

, where

[x]

is the greatest integer not greater than

x

, then

f^{'} (1^{+})

= ….

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

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?

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

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:

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

f (x) = \{\begin{matrix} x & for & x \leq 0 \\ 0 & for & x > 0 \end{matrix}

then

f (x)

x = 0

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

Let

f : R \to R

be a function defined by

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

. Let

S

denotes the set of all points in

R

, where

f

is not differentiable. Then

EASY

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

Two sentences are given below and you are required to find the correct sentence which combines both the sentences.

Which is the correct combination of the given two sentences?

He is too tired. He could not stand.

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

Let

f : (- 1, 1) \to R

be a function defined by

f (x) = m a x \{- |x|, - \sqrt{1 - x^{2}}\} .

K

be the set of all points at which

f

is not differentiable, then

K

has exactly

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

In the following sentence, a blank space with four options is given. Select whichever preposition or determiner you consider the most appropriate for the blank space.

These are the good rules to live _____.

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

Let

S = {(λ, μ) \in R \times R : f (t) = (|λ| e^{|t|} - μ) \sin (2 |t|), t \in R

is a differential function}. Then,

S

is a subset of :

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

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:

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

2 y = {(\cot^{- 1} (\frac{\sqrt{3} \cos x + \sin x}{\cos x - \sqrt{3} \sin x}))}^{2} \forall x \in (0, \frac{π}{2})

, then

\frac{d y}{d x}

is equal to

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

The function

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

is :

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

Let a function,

f : (- 1, 3) \to R

be defined as

f (x) = \min {x [x], | x [x] - 2 | + 2},

where

[x]

denotes the greatest integer

\leq x .

Then

f

is :

HARD

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

The function

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

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

f (x) = \{\begin{matrix} x e^{- (\frac{1}{|x|} + \frac{1}{x})}; & i f & x \neq 0 \\ 0; & i f & x = 0 \end{matrix}

then which of the following is correct?

MEDIUM

Mathematics>Differential Calculus>Continuity and Differentiability>Differentiability

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>Differential Calculus>Continuity and Differentiability>Differentiability

f (x) = \{\begin{matrix} x \sin (\frac{1}{x}), & x \neq 0 \\ 0, & x = 0 \end{matrix}

x = 0

Let f:R→R be defined as fx=a−bcos2xx2;x<0x2+cx+2;0≤x≤12x+1;x>1 If f is continuous everywhere in R and m is the number of points where f is NOT differential then m + a + b + c equals:

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

Let $f : R \to R$ be defined as

$f (x) = \{\begin{matrix} \frac{a - b \cos 2 x}{x^{2}}; x < 0 \\ x^{2} + c x + 2; 0 \leq x \leq 1 \\ 2 x + 1; x > 1 \end{matrix}$

If $f$ is continuous everywhere in $R$ and $m$ is the number of points where $f$ is NOT differential then $m + a + b + c$ equals: