$.$

$($a$)$ State the Inverse Function Theorem for vector functions.

$($b$)$ Show that the function $G$:$R^{3}longrightarrow{R}^3$ given by

$G(x,y,z)=(x+y+z^2,x-y+z,2x+y-z)$

has a local differentiable inverse around the point $(4,4,-1).$ Compute $D{G}^{-1}(4,4,-1).$

$($c$)$ Does the function $G$ from above admit a global inverse?