Question

Let f(x) = x | x | and g(x) = sin x.

Statement-1: gof is differentiable at x = 0 and its derivative is continuous at that point.

Statement-2: gof is twice differentiable at x = 0.

(A) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1

(B) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1

(C) Statement-1 is true, Statement-2 is false

(D) Statement-1 is false, Statement-2 is true

Answer 1

Answer is (C) Statement-1 is true, Statement-2 is fals

We have

Clearly, L(gof)′ (0) = 0 = R(gof)′ (0)

Therefore, gof is differentiable at x =  0 and also its derivative is continuous at x = 0.

Now,

L(gof)′′ (0) = -2 and R(gof)′′ (0) = 2.

Hence L(gof)′′ (0) ≠ R(gof)′′ (0)

That is, gof(x) is not twice differentiable at x = 0.