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

Last Updated : 16 May, 2021

Question 1. Show that f(x) = |x â€“ 3| is continuous but not differentiable at x = 3.

Solution:

f(3) = 3 â€“ 3 = 0

=

= 0

= 0

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

Now,

= â€“1

= 1

Since (LHD at x = 3) â‰  (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 â€“ 2 = 6

= 8 â€“ 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) =

= â€“1

Clearly LHD â‰  RHD at x = 1

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

(LHD at x = 2) =

= â€“1

(RHD at x = 2) =

= â€“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 Ã— k

= 0

(RHD at x = 0)

=  0 Ã— k

= 0

Clearly LHL = RHL at x = 0

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

Question 7(ii) Show thatis 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 thatis not differentiable at x = 0, if mâ‰¤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 â€“ 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) =

= â€“2

Since (LHD at x = 1) â‰  (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 â€“ b = 1                        …..(1)

(LHD at x = 1) =

Using (1), we get

= 2a

(RHD at x =1)

= â€“1

Since f(x) is differentiable, LHL = RHL

or, 2a = â€“1

a = â€“1/2

Substituting a = â€“1/2 in (1), we get,

b = â€“1/2 â€“ 1

b = â€“3/2

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