Second partial derivatives of harmonic functions

Question

Given a twice-differentiable function $f:S\rightarrow \mathbb{R}$, where $S$ is a nonempty subset of $\mathbb{R}^n$, how does one prove that if f is harmonic then the second partials $D_jD_kf$ all vanish at $(x,y) = (x_0,y_0)$, where $x_0$ and $y_0$ are some local minima or maxima of $f$ in $S$?