I was looking at an old Berkeley preliminary exam problem (Fall,88) stated below;
Prove that a real-valued $C^3$ function $f$ on $\mathbb{R}^2$ whose Laplacian
$\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}$
is everywhere positive cannot have a local maximum.
[End of question]
In the solution, it is mention that "... since $f\in C^3$, for $f$ to have a relative maximum its Hessian must have negative eigenvalues ..."
What I know is that if Hessian has negative eigenvalue, then it is negative definite and hence the critical point is a local maximum, but is the converse used above true? What is its justification?
I hope someone can shade a light on this, thanks.