Prove that if $\frac{\partial^2 f}{\partial x^2}(\textbf{a})\neq 0$ then $\textbf{a}$ cannot be extrimum of $f$

Question

Let $f:\mathbb{R^2} \to \mathbb{R}$ in $C^2$ and harmonic. If $\frac{\partial^2 f}{\partial x^2}(\textbf{a})\neq 0$ then $\textbf{a}$ cannot be extrimum of $f$

i am not getting any idea..can someone help me with this problem

Answer 1

If $f$ is harmonic, then the trace of the Hessian matrix is zero. The trace is equal to the sum of the two eigenvalues, which are real. Since the Hessian is not zero at $a$, it must have a non-zero eigenvalue. Both properties together imply that the Hessian has one negative eigenvalue and one positive eigenvalue.