Let $f$ be a function with continuous second derivative and $f\left(0\right)=0, f^{\text{'}}\left(0\right)=0.$ Define a function $g$ by 

$g\left(x\right)=\left\{\begin{array}{ll}\frac{f\left(x\right)}{x},& x\ne 0\\ 0,& x=0\end{array}\right.$.
Then, which of the following statements is correct?

Let $f\left(x\right)$ be a function defined on $(-a, a)$ with $a>0$. Assume that $f\left(x\right)$ is continuous at $x=0$ and $\underset{x\to 0}{\lim }\frac{f\left(x\right)-f\left(kx\right)}{x}=\alpha$,
where $k\in (0,1)$, then

If $f\left(x\right)=x^{\frac{1}{3}}{\left(x-2\right)}^{\frac{2}{3}}$ for all $x,$ then the domain of $f^{\text{'}}$ is: