Consider the map $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ given by

$$f(x, y)=\left(x^{1 / 3}+y^{2}, y^{5}\right)$$

where $x^{1 / 3}$ denotes the unique real cube root of $x \in \mathbb{R}$.

(a) At what points is $f$ continuously differentiable? Calculate its derivative there.

(b) Show that $f$ has a local differentiable inverse near any $(x, y)$ with $x y \neq 0$.

You should justify your answers, stating accurately any results that you require.