Related Articles

# Class 12 RD Sharma Solutions – Chapter 10 Differentiability – Exercise 10.1

• Last Updated : 16 May, 2021

### Question 1. Show that f(x) = |x – 3| is continuous but not differentiable at x = 3.

Solution: f(3) = 3 – 3 = 0  = = 0   = 0

Since LHL = RHL, f(x) is continuous at x = 3.

Now,  = –1  = 1

Since (LHD at x = 3) ≠ (RHD at x = 3)

f(x) is continuous but not differentiable at x =3.

### Question 2. Show that f (x) = x1/3 is not differentiable at x = 0.

Solution:

(LHD at x = 0) =   = Undefined

(RHD at x = 0) =   = Undefined

Clearly LHD and RHD do not exist at 0.

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

### Question 3. Show that is differentiable at x = 3.

Solution:

(LHD at x = 3) =  = 12

RHD at x = 3 =  = 12

Since LHL = RHL

f(x) is differentiable at x = 3.

### Question 4. Show that the function f is defined as follows is continuous at x = 2, but not differentiable thereat: Solution:

f(2) = 2(2)2 – 2 = 6  = 8 – 2

= 6   = 6

Clearly LHL = RHL at x = 2

Hence f(x) is differentiable at x = 2.

### Question 5. Discuss the continuity and differentiability of the function f(x) = |x| + |x -1| in the interval of (-1, 2).

Solution:  (LHD at x = 0) =  = 2

(RHD at x = 0) =  = 0

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

### Question 6. Find whether the following function is differentiable at x = 1 and x = 2 or not. Solution:

(LHD at x = 1) =  = 1

(RHD at x = 1) =  = –1

Clearly LHD ≠ RHD at x = 1

So f(x) is not differentiable at x = 1.

(LHD at x = 2) =  = –1

(RHD at x = 2) =  = –1

Clearly LHL = RHL at x = 2

Hence f(x) is differentiable at x = 2.

### Question 7(i). Show that is differentiable at x = 0, if m>1.

Solution:

(LHD at x = 0) =   = 0 × k

= 0

(RHD at x = 0)   =  0 × k

= 0

Clearly LHL = RHL at x = 0

Hence f(x) is differentiable at x = 0.

### Question 7(ii) Show that is not differentiable at x = 0, if 0<m<1.

Solution:

(LHD at x = 0)   = Not defined

(RHD at x = 0)   = Not defined

Clearly f(x) is not differentiable at x = 0.

### Question 7(iii)Show that is not differentiable at x = 0, if m≤0.

Solution:

(LHD at x = 0)   = Not defined

(RHD at x = 0)   = Not defined

Clearly f(x) is not differentiable at x = 0.

### Question 8. Find the value of a and b so that the function is differentiable at each real value of x.

Solution:

(LHD at x = 1) =  = 5

(RHD at x = 2) =  = b

Since f(x) is differentiable at x = 1,so

b = 5

Hence, 4 + a = b + 2

or, a = 7 – 4 = 3

Hence, a = 3 and b = 5.

### Question 9. Show that the function is notdifferentiable at x =1.

Solution:

(LHD at x = 1) =   = 0

(RHD at x =1) =   = –2

Since (LHD at x = 1) ≠ (RHD at x = 1)

f(x) is continuous but not differentiable at x =1.

#### Question 10. If is differentiable at x = 1, find a and b.

Solution:

We know f(x) is continuous at x = 1.

So, a – b = 1                        …..(1)

(LHD at x = 1) =  Using (1), we get = 2a

(RHD at x =1)  = –1

Since f(x) is differentiable, LHL = RHL

or, 2a = –1

a = –1/2

Substituting a = –1/2 in (1), we get,

b = –1/2 – 1

b = –3/2

Attention reader! All those who say programming isn’t for kids, just haven’t met the right mentors yet. Join the  Demo Class for First Step to Coding Coursespecifically designed for students of class 8 to 12.

The students will get to learn more about the world of programming in these free classes which will definitely help them in making a wise career choice in the future.

My Personal Notes arrow_drop_up