if $g$ be a continuous function not differentiable at $0$

Question

Let $g$ be a continuous function not differentiable at $0$ with $g(0)=8$. Let $f(x)=x\,g(x)$ .Find $f'(0)$

a)$0$
b)$4$
c)$2$
d)$8$

I am getting that $f'(x)=g(x)+x\,g'(x)$. But since $g'(x)$ doesn't exist for $x=0$, hence $f'(x)=8$. Please help whether it is right or wrong

Answer 1

Guide:

\begin{align} f'(0) &= \lim_{h \to 0} \frac{f(h)-f(0)}{h} \\ &= \lim_{h \to 0}\frac{f(h) - 0 \cdot g(0)}{h} \\ &= \lim_{h \to 0} \frac{f(h)}{h} \end{align}

Can you complete the computation above?

Answer 2

It's wrong. You should try from the definition of differentiability.

Hint: $f^{'}(0)=\lim_{h \to 0} \frac{f(0+h)-f(0)}{h}$.

Answer 3

$\lim_{x \rightarrow 0} \dfrac{f(x)-f(0)}{x} =$

$\lim_{x \rightarrow 0} g(x) = g(0)$, since

$g$ is continous at $0$.

Note: $g'(0)$ does not exist .

Which answer?