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# 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

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