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

**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) = x**^{1/3} is not differentiable at x = 0.

^{1/3}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 = –1/2Substituting a = –1/2 in (1), we get,

b = –1/2 – 1

b = –3/2

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