Question

Let g:R → R be a differential function g(0) = 0, g'(0) = 0 and g'(1) ≠ 0

Let f(x) = {(xg(x)/|x| : x ≠ 0), (0 : x = 0)

and h (x) = e^|x| for all x in R

Let (f₀h)(x) denote f(h(x))

and (h₀f)(x)denote h( f(x))

Then which of the following is (are) true?

(a) f is differentiable at x = 0

(b) h is differentiable at x = 0

(c) (f₀h) is differentiable at x = 0

(d) (h₀f) is differentiable at x = 0

Answer 1

Correct option (a, b, c, d)

Explanation:

Differentiability at x = 0

Thus, f(x) is differentiable at x = 0

Differentiability of h (x) at x = 0

h'(0⁺) = 1, h(x) is an even function.

Hence, non differentiable at x = 0

Differentiability of f(h (x)) at x = 0

since g'(1) ≠ 0, so f(h(x)) is non differentiable at x = 

Differentiability of h ( f(x)) at x = 0