Let f:R→R and g:R→R be functions satisfying f(x + y) = f(x) + f(y) + f(x) f(y) and f(x) = xg(x) for all x,y ∈ R. If limx → 0 g(x) = 1

Question

Let f:R→R and g:R→R be functions satisfying f(x + y) = f(x) + f(y) + f(x) f(y) and f(x) = xg(x) for all x,y ∈ R. If \( \lim\limits_{x \to 0}\) g(x) = 1, then which of the following statements is/are TRUE?

(A) f is differentiable at every x ∈ R

(B) If g(0) = 1, then g is differentiable at every x ∈ R 

(C) The derivative f′(1) is equal to 1 

(D) The derivative f'(0) is equal to 1

Answer 1

(A) f is differentiable at every x ∈ R,

(B) If g(0) = 1, then g is differentiable at every x ∈ R,

(D) The derivative f'(0) is equal to 1

since f(x) = xg(x)

Using partial derivative (w.r.t. y) 

∴ f(x) = e^x – 1 

Option (A) is correct 

And f'(x) = e^x 

f'(0) = 1 option(D) is correct

Option B is correct