The question and its solution are given below:
Find for which points in the $xy-plane$ the system $x = u + v, y= u^2 + v^2, z = u^3 + 2v^3 $ determine $z$ as a differentiable function of $x$ and $y$.
The solution is :
1-But I can not understand how the solution uses the implicit function theorem ? or may be I do not understand exactly which theorem the solution uses.
2-Also why $2(u^2 + v^2) - (u + v)^2 \geq 0?$
3- why we must have $u,v$ as functions of $x,y$?