Let f : R → R be a function satisfying $f(x+y)=f(x)+f(y)$ for all $x, y \in R$. If $f(x)=x^3 g(x)$ for all $x, y \in R$, where g(x) is continuous, then f'(x) is equal to

0

We have,

$f'(x) =\lim\limits_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$

$\Rightarrow f'(x) =\lim\limits_{h \rightarrow 0} \frac{f(x)+f(h)-f(x)}{h}$         [∵ f(x + y) = f(x) + f(y)]

$\Rightarrow f'(x) =\lim\limits_{h \rightarrow 0} \frac{f(h)}{h}$

$\Rightarrow f'(x)=\lim\limits_{h \rightarrow 0} \frac{h^3 g(h)}{h}$

$\Rightarrow f'(x)=\lim\limits_{h \rightarrow 0} h^2 \times \lim\limits_{h \rightarrow 0} g(h) $

$\Rightarrow f'(x)=0 \times g(0)$         [∵ g is continuous   ∴ $\lim\limits_{h \rightarrow 0} g(h)=g(0)]$

$\Rightarrow f'(x)=0$