Let $f : \mathbb{R}^n \rightarrow \mathbb{R}, f \in C^2$. I have to prove that for every nondegenerate critical point of $f$, there exists a neighbourhood which does not contain any further critical points.
I know that I will have to look at the Taylor formula and I'm pretty sure that I have to use that $\det (\partial_{ij} f(x)) \neq 0$ for all $x$ which are nondegenerate, that is to say the Jacobi-matrix is invertible. This makes me feel like I should use the Inverse function theorem, but to what end?
I'd be really delighted if I could get some advice. Thanks in advance.