Suppose a function $f(x,y)$ is defined and differentiable anywhere in $\mathbb{R}^2$. Suppose also $f(0,0)$ is a local maximum with second-derivative test (namely, locally concave down, or, negative-definite Hessian). Of course we know that the gradient at $(0,0)$ is zero, namely, a critical point. Furthermore, suppose that the origin is the only critical point. Prove or disprove that the origin the only global maximum.
My intuition is “proof”. Note that by Fermat’s theorem on stationary points (i.e. critical points), any other global maximum have to occur on the boundary of $\mathbb{R}^2$, namely at infinity. But that’s where I got stuck. For example, if I find a value same as that of origin $f(x,y) =f(0,0)$ and I use the usual idea of invoking Rolle Theorem or Taylor theorem I can only get a particular directional gradient being 0, and not the gradient vector itself being 0. Am I on the right track? Any other ideas? Thanks!