Prove that if a function has a Hessian with zeroes on the diagonal, the function doesn't have local maxima

Question

For a function who's Hessian in the form $$\begin{pmatrix} 0 & a & b\\ a & 0 & c\\ b & c & 0\\ \end{pmatrix}$$ prove that the function can not have any local maxima.

I tried to prove this by looking at the determinants of the Hessian to determine the extrema of the function. However, I did not get very far.