Let Ω be a nonempty open and bounded subset of $\mathbb R^n$ and let $f\colon \barΩ \to \mathbb R$ $ $ ($ \barΩ$ is the closure of Ω) be a function that is continuous on $\barΩ$ and differentiable on Ω. If the function f is constant on $∂Ω$ (boundary of Ω), prove that f has a critical point in Ω.
My first reasoning was the following: f is defined on a compact domain and it is continous on this set, then by the extreme value theorem f has an extreme point in $\barΩ$. If I can show that this extreme point is not on the boundary $∂Ω$, then it has to be in the interior, and by the fact that f is differentiable in the interior, gradient at that point must be a zero vector (so we have our critical point).
But I have stuck in showing that for any x$\in$$∂Ω$, there is another $\acute{x}$$\in$$Ω$ such that f($\acute{x}$)$\gt$f(x), or f($\acute{x}$)$\lt$f(x).
I would appreciate any help in proving this theorem, thanks.