Local minimum implies global minimum.

Question

Let $f \in C^1(\mathbb{R}^n, \mathbb{R})$ (or even smoother if that helps) with $n > 1$. Assume that $f$ has a local minimum in $x_0 \in \mathbb{R}^n$ and no other stationary points. Is $x_0$ then automatically a global minimum? For $n=1$ this is true and easy to show (e.g. with Rolle's theorem)...
My intuition says the answer should be yes but I can't prove it.