Using the implicit function theorem one can prove the following:
Let $X,Y$ be Banach-spaces, $U\subset X$ open, $f\colon U\to \mathbf{R}$, $g\colon U\to Y$ continuously differentiable function. If $f|_{g^{-1}(0)}$ has a local extremum at $x$, and g'(x)\in L(X;Y) has a right inverse in $L(Y;X)$, then there is a unique $\lambda\in Y^*$, such that f'(x)=\lambda\circ g'(x).
One can also give some second order necessary and sufficient conditions. This generalizes the case where $Y$ is finite dimensional, and g'(x) is surjective. However I have seen some texts that claimed that only surjectivity is sufficient even in the infinite dimensional case.
My question is does that hold true? If it does what is the method of the proof, because I cannot figure how the implicit function theorem could be used. If the proof is complicated I would appreciate even just some good references.