Where $\sum p_i = 1$.
I have to show that $f(x)=g(x)-u(x)$ is always negative or $0$ over $\Bbb R_+^n$. I've already shown that $f$ has critical points all the way along the diagonal $\Bbb R(1, ... 1)$, that it is $0$ at each of these critical points, and that its Hessian at each of these points is negative definite, so I know that they're all local maximums. But to conclude what I need, I have to show that they're also global maxima.
How can I do this? I know there are no other maxima (because there are no other critical points), but the function might tend to something positive at infinity, for example. Should I study the behavior of $f(x)$ as $|x|\to\infty$? How would I approach that for a multivariable function like this?