Q:

Confusing fact about minimum and maximum

Let $f(x)$ and $g(x)$ be functions. I'm looking for an upper bound on $-\min(-f(x), -g(x))$, but the minus signs are making me dizzy. Is this the same as an upper bound on $\max(f(x),g(x))$?

A:

Pick a number $K$.
$$ -\operatorname{min}(-f,-g) \leq K \iff \operatorname{min}(-f,-g) \geq -K $$
Using Srivatsan's suggestion, this is
$$ -\operatorname{max}(f,g) \geq -K $$
which is equivalent to $\operatorname{max}(f,g) \leq K$.  So you were right!

