The statement that maximizing a function over its argument is equivalent to minimizing that function over the same argument with a sign change seems to be accepted as trivial wherever I look (MSE, proofwiki, internet search, textbooks outside of optimization theory). Intuitively, if you have some function of a single variable that has a global maximum, and you "flip it over" by changing the sign, the global maximum is now a global minimum. I can appreciate that.
However, it seems to me that math is all about meticulous examination of surprising subtleties. Does anyone know of a good way to prove this statement? I just don't feel comfortable without it.