The following system of three nonlinear algebraic equations is to be solved for $~x, y, z~ $ as functions of the variables $~u, v,w~$:
$u = x + y + z$
$v = x^2 + y^2 + z^2$
$w = x^3 + y^3 + z^3$
$a)$ Prove or find a counter example: for each $(u,v,w)$ near $(0,2,0)$, there is a unique solution $(x,y,z)$ near $(-1,0, 1)$.
$b)$ Is the Implicit Function Theorem applicable for $(u,v,w)$ near $(2, 4, 8)$ and $(x,y, z)$ near $(0, 0, 2)$?