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

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

Let f : R → R be given by f x = x 5 + 5 x 4 + 10 x 3 + 10 x 2 + 3 x + 1 , x < 0 ; x 2 - x + 1 , 0 ≤ x < 1 ; 2 3 x 3 - 4 x 2 + 7 x - 8 3 , 1 ≤ x < 3 ; x - 2 l o g e x - 2 - x + 10 3 , x ≥ 3 Then which of the following options is/are correct?

f ' has a local maximum at x = 1

For every twice differentiable function f : ℝ → - 2 , 2 with f 0 2 + f ′ 0 2 = 85 , which of the following statement(s) is (are) TRUE?

There exists α ∈ - 4 , 4 such that f α + f ′ ′ α = 0 a n d f ′ α ≠ 0

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

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

Let a , b ∈ R and f : R → R be defined by f x = a cos ⁡ x 3 - x + b x sin ⁡ x 3 + x . Then f is -

Differentiable at x = 1 if a = 1 and b = 0

NOT differentiable at x = 0 if a = 1 and b = 0

NOT differentiable at x = 1 if a = 1 and b = 1

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

Let the function f : R → R be defined by f x = x 3 − x 2 + x − 1 sin x and let g : R → R be an arbitrary function. Let f g : R → R be the product function defined by f g x = f x g x . Then which of the following statements is/are TRUE?

If g is differentiable at x = 1 , then f g is differentiable at x = 1

If f g is differentiable at x = 1 , then g is continuous at x = 1

If f g is differentiable at x = 1 , then g is differentiable at x = 1

Let f : R → R be a function such that f x ≤ x 2 , for all x ∈ R . Then, at x = 0 , f is

continuous as well as differentiable

differentiable but not continuous

neither continuous nor differentiable

continuous but not differentiable

HARD

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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 $ϕ$ is:

50% studentsanswered this correctly

Important Questions on Differentiation

HARD

Mathematics>Differential Calculus>Differentiation>Derivative of a Function

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:

HARD

Mathematics>Differential Calculus>Differentiation>Derivative of a Function

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>Differentiation>Derivative of a Function

Let

f : R \to R

be given by

f (x)

= \{\begin{matrix} x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 3 x + 1, & x < 0; \\ x^{2} - x + 1, & 0 \leq x < 1; \\ \frac{2}{3} x^{3} - 4 x^{2} + 7 x - \frac{8}{3}, & 1 \leq x < 3; \\ (x - 2) {l o g}_{e} (x - 2) - x + \frac{10}{3}, & x \geq 3 \end{matrix}

Then which of the following options is/are correct?

HARD

Mathematics>Differential Calculus>Differentiation>Derivative of a 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:

EASY

Mathematics>Differential Calculus>Differentiation>Derivative of a Function

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?

HARD

Mathematics>Differential Calculus>Differentiation>Derivative of a Function

f (x) = [x]

, where

[x]

is the greatest integer not greater than

x

, then

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

= ….

HARD

Mathematics>Differential Calculus>Differentiation>Derivative of a Function

If the function

f (x) = \{\begin{matrix} - x, & x < 1 \\ a + \cos^{- 1} (x + b), & 1 \leq x \leq 2 \end{matrix}

is differentiable at

x = 1

, then

\frac{a}{b}

is equal to

EASY

Mathematics>Differential Calculus>Differentiation>Derivative of a Function

At how many points between the interval

(- \infty, \infty)

, is the function

f (x) = \sin x

not differentiable

HARD

Mathematics>Differential Calculus>Differentiation>Derivative of a Function

For every twice differentiable function

f : ℝ \to [- 2, 2]

with

{(f (0))}^{2} + {(f^{'} (0))}^{2} = 85

, which of the following statement(s) is (are) TRUE?

MEDIUM

Mathematics>Differential Calculus>Differentiation>Derivative of a Function

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?

HARD

Mathematics>Differential Calculus>Differentiation>Derivative of a Function

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 :

HARD

Mathematics>Differential Calculus>Differentiation>Derivative of a Function

Let

a, b \in R

and

f : R \to R

be defined by

f (x) = a \cos (|x^{3} - x|) + b |x| \sin (|x^{3} + x|)

. Then

f

is -

MEDIUM

Mathematics>Differential Calculus>Differentiation>Derivative of a Function

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

MEDIUM

Mathematics>Differential Calculus>Differentiation>Derivative of a Function

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:

HARD

Mathematics>Differential Calculus>Differentiation>Derivative of a Function

Let the function

f : R \to R

be defined by

f (x) = x^{3} - x^{2} + (x - 1) \sin x

and let

g : R \to R

be an arbitrary function. Let

f g : R \to R

be the product function defined by

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

. Then which of the following statements is/are TRUE?

EASY

Mathematics>Differential Calculus>Differentiation>Derivative of a Function

Consider the greatest integer function, defined by

f (x) = [x], 0 \leq x < 2

. Then

MEDIUM

Mathematics>Differential Calculus>Differentiation>Derivative of a Function

f (x) = \{\begin{array}{cl} e^{x}, & x \leq 0 \\ |1 - x|, & x > 0 \end{array}

, then

HARD

Mathematics>Differential Calculus>Differentiation>Derivative of a Function

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

MEDIUM

Mathematics>Differential Calculus>Differentiation>Derivative of a 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>Differential Calculus>Differentiation>Derivative of a 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

For each x∈R, let fx=x-1, gx=cos⁡x and ϕx=fg2sin⁡x-gfx. Then ϕ is:

Important Questions on Differentiation

Learn and Prepare for any exam you want !

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 $ϕ$ is: