Sol. We have,

f(x) =

Clearly, it is continuous and differentiable for all x except at x = 0 and x = 1. So, we check the differentiability at x = 0 and 1.

Now,

(LHD at x = 0)

=

=

=

=

=

= h sin + 2 sin + 1 + = 0 × sin 1 + 2 sin 1 + 1 +

= 2 sin 1 + 1 + 2

= 2 sin 1 + 1 –2

= 2 sin 1 + 1 – 2

= 2 sin 1 + 1 –cos 1

(RHD at x = 0)

=

=

=

= h sin –2 sin –1 +

= 2 sin 1 –1 +

= 2 sin 1 –1 + 2

= 2 sin 1 –1 – cos 1

Clearly, (LHD at x = 0) ≠ (RHD at x =0)

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

Now,

(LHD at x = 1)

=

=

=

=

= h sin –1 = –1

and,

(RHD at x = 1)

=

=

=

=

= h sin –1 = –1

∴ (LHD at x = 1) = (RHD at x = 1)

So, f(x) is differentiable at x =1

Hence, f(x) is differentiable for all x, except at x = 0