Let $f:\mathbb{R}^2\to \mathbb{R}$ be of class $C^1$, with $f(2,-1)=-1$. Set $G(x,y,u)=f(x,y)+u^2, H(x,y,u)= ux+3y^3+u^3$.
(a) What conditions on $Df$ ensure that there are $C^1$ functions $x = g(y)$ and $u = h(y)$ defined on an open set in $\mathbb{R}$ that satisfy both equations, such that $g(−1) = 2$ and $h(−1) = 1$?
(b) Under the conditions of (a), and assuming that $Df(2, -1) = [1 -3]$, find $g'(-1)$ and $h'(-1)$.
There is already a post about this here What conditions on a function's derivative will satisfy given conditions? but it does not seem very clear to me.
The truth is that I am very confused about this problem, could someone explain to me in detail what should I do?
For example for (a) I have thought that you can apply the implicit function theorem to $f$ to find $g$, the problem is that $f(2,-1)=-1\neq 0$ and I need $f(2,-1)=0$, how could I do this? How do I do with the other questions? Thank you very much.