Relationship between maximum and minimum of a function

Question

Does $\max(f) = -\min(-f)$ hold generally?

Actually, it is $\max(f)=-\min(-f)$, provided at least one of them exists. — sranthrop, Jan 06 '15 at 14:01
@nomadStack: And now you know the answer. Dou you want to see a proof? :) — sranthrop, Jan 06 '15 at 14:12
@sranthrop: Well, I'm not sure. I'm doing some optimization with my computer and I though, why not just use minimize algorithms like the Nelder-Mead on the product of the function with -1 instead of maximizing? I'm neither sure much this holds computationally, let alone mathematically. — nomadStack, Jan 06 '15 at 14:15

score 1 · Accepted Answer · answered Jan 06 '15 at 14:19

1

Let's assume that $\max(f)=m$. Then there exists some $x_0$ such that $f(x_0)=m$ and $f(x)\leq m$ for each $x$. This implies $-f(x)\geq -m$ and $-f(x_0)=-m$, so $\min(-f)=-m$.

answered Jan 06 '15 at 14:19

sranthrop

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Relationship between maximum and minimum of a function

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