# RD SHARMA Solutions for Class 12-science Maths Chapter 10 - Differentiability

## Chapter 10 - Differentiability Exercise MCQ

The function f: R → R given by f(x) = |x - 1| is

a. continuous as well as differentiable at x = 1

b. not continuous but differentiable at x = 1

c. continuous but not differentiable at x = 1

d. neither continuous nor differentiable at x = 1

Therefore, f(x) is continuous at x = 1.

Now,

Therefore, Lf'(1) ≠ Rf'(1).

Hence, f(x) is not differentiable at x = 1.

The function is

a. continuous everywhere but not differentiable at x = 0

b. continuous and differentiable everywhere

c. not continuous at x = 0

d. none of these

Given:

Now, f(x) is continuous everywhere except at x = 0.

Therefore, f(x) is continuous everywhere.

Now,

Therefore, Lf'(1) ≠ Rf'(1).

Thus, f(x) is not differentiable at x = 0.

Hence, f(x) is continuous everywhere but not differentiable at x = 0.

The set of points where the function is differentiable, is

a. R

b.

c. (0, ∞)

d. none of these

Given:

Now,

Therefore, Lf'(1) ≠ Rf'(1).

Thus, f(x) is not differentiable at

Hence, the set of points where f(x) is differentiable is

Let f(x) = |x| and g(x) = |x^{3}|, then

- f(x) and g(x) both are continuous at x = 0
- f(x) and g(x) both are differentiable at x = 0
- f(x) is differentiable but g(x) is not differentiable at x = 0
- f(x) and g(x) both are not differentiable at x = 0

Correct option: (a)

Absolute value function is continuous on R.

The function f(x) = sin^{-1} (cos x) is

- discontinuous at x = 0
- continuous at x = 0
- differentiable at x = 0
- none of these

Correct option: (b)

The set of points where the function f(x) = x|x| is differentiable is

- (-∞, ∞)
- (-∞, 0) ∪ (0, ∞)
- (0, ∞)
- [0, ∞]

Correct option: (a)

- continuous at x = -2
- not continuous at x = -2
- differentiable at x = -2
- continuous but not derivable at x = -2

Correct option: (b)

Let f(x) = (x + |x|) |x|. Then, for all x

- f is continuous
- f is differentiable for some x
- f ' is continuous
- f" is continuous

Correct option: (a), (c)

The function f (x) = e^{-|x|} is

- continuous everywhere but not differentiable at x =0
- continuous and differentiable everywhere
- not continuous at x =0
- none of these

Correct option: (a)

- continuous on [-1,1] and differentiable on (-1,1)
- continuous on [-1,1] and differentiable (-1,0)∪ ϕ(0,1)
- continuous and differentiable on [-1,1]
- none of these

Correct option: (b)

If f (x) = a |sin x| +b e^{|x|}+c|x|^{3} and if f (x) is differentiable at x =0,then

- a = b = c = 0
- a = 0, b = 0; c ∊ R
- b = c = 0; a ∊ R
- c = 0 , a = 0 , b ∊ R

Correct option: (b)

- has no limit
- is discontinuous
- is continuous but not differentiable
- is differentiable

Correct option: (b)

If f (x)= |log_{e }x|, then

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

Correct option: (a), (b)

If f (x) = log_{e} |x|, then

- f(x) is continuous and differentiable for all x in its domain
- f (x) is continuous for all for all x in its domain but not differentiable at x = ± 1
- f (x) is neither continuous nor differentiable at x = ± 1
- none of these

Correct option: (b)

Correct option: (b)

The function f(x) =x -[x], where [.] denotes the greatest integer function is

- continuous everywhere
- continuous at integer point only
- continuous at non-integer points only
- differentiable everywhere

Correct option: (c)

- a = 2
- a = 1
- a =0
- a =1/2

Correct option: (d)

Let f (x) =|sin x|. Then,

- f (x) is everywhere differentiable.
- f (x) everywhere continuous but not differentiable at x = n π, n ∊ Z

- None of these

Correct option: (b)

Let f (x) =|cos x|. Then,

- f (x) is everywhere differentiable
- f (x) everywhere continuous but not differentiable at x = n π, n ∊ Z

- None of these

Correct option: (c)

The function f (x) =1+|cos x| is

- Continuous no where
- Continuous everywhere
- Not differentiable at x =0
- Not differentiable at x =n π, n ∊ Z

Correct option: (b)

The function f (x) = |cos x| is

- Differentiable at x = (2n+1) π/2, n∊ Z
- Continuous but not differentiable at x =(2n+1) π/2, n ∊ Z
- Neither differentiable nor continuous at x = n π, n ∊ Z
- None of these

Correct option: (b)

- Continuous as well differentiable for all x ∊ R
- Continuous for all x but not continuous at some x.
- Differentiable for all but not continuous at same x'.
- None of these

Correct option: (a)

Let f (x) = ɑ + b |x|+c|x|^{4}, where ɑ, b and c are real constants. Then f (x) is differentiable at x =0 , if

- a =0
- b =0
- c =0
- none of these

Correct option: (b)

If f (x) = |3-x|+(3+x) , where (x) denotes the least integer greater than or equal to x, then f (x) is

- continuous and differentiable at x =3
- continuous but not differentiable at x = 3
- differentiable but not continuous at x =3
- neither differentiable nor continuous at x =3

Correct option: (d)

- continuous as well as differentiable at x =0
- continuous but not differentiable at x = 0
- differentiable but not continuous at x =0
- none of these

Correct option: (d)

- continuous and differentiable
- differentiable but not continuous
- continuous but not differentiable
- neither continuous nor differentiable

Correct option: (a)

The set of point where the function f (x) = |x-3| cos x is differentiable

- R
- R - {3}
- (0,∞ )
- None of these

Correct option: (b)

- Continuous at x =1
- Differentiable at x = -1
- Everywhere continuous
- Everywhere differentiable

Correct option: (b)

## Chapter 10 - Differentiability Exercise Ex. 10.1

Find whether the following functions is differentiable at

x = 1 and x = 2 or not:

Find the values of a and b, if the function f(x) defined by is differentiable at x = 1.

Given:

As f(x) is differentiable at x = 1, we have

… (i)

Now, Rf'(1) exist when (b - 2 - a) = 0 … (ii)

From (i), we get, b = 5

Putting this in (ii), we get, a = 3.

## Chapter 10 - Differentiability Exercise Ex. 10.2

Examine the differentiability of the function f defined by

## Chapter 10 - Differentiability Exercise Ex. 10VSAQ

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