Show that the map $f: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ given by

$$f(x, y, z)=\left(x-y-z, x^{2}+y^{2}+z^{2}, x y z\right)$$

is differentiable everywhere and find its derivative.

Stating accurately any theorem that you require, show that $f$ has a differentiable local inverse at a point $(x, y, z)$ if and only if

$$(x+y)(x+z)(y-z) \neq 0 .$$

$$\begin{aligned} & p(f)=\sup |f|, \quad q(f)=\sup \left(|f|+\left|f^{\prime}\right|\right), \\ & r(f)=\sup \left|f^{\prime}\right|, \quad s(f)=\left|\int_{-1}^{1} f(x) d x\right| \end{aligned}$$