Show that $f(x_1^*,...x_n^*)=\max\{f(x_1,...,x_n):(x_1,...,x_n)\in\Omega\}$ if and only if $-f(x_1^*,...x_n^*)=\min\{-f(x_1,...,x_n):(x_1,...,x_n)\in\Omega\}$
I am not exactly sure how to approach this problem -- it is very general, so I can't assume anything about the shape of $f$. It seems obvious that flipping the $\max$ problem with a negative turns it into a $\min$ problem. Thoughts?