The following fairly obvious theorem is surprisingly commonly used:
Let $f: [0;+\infty)\mapsto [0;+\infty)$ be (right) differentiable at $0$ and $f'(0)>0$, $\lim_{x\rightarrow+\infty} f(x) = 0$, then $f$ reaches its maximum value in the interior of its domain, i.e., $\exists m>0: x\ge0 \implies f(x)\le f(m)$.
Does it have an official name?
PS. Proof:
$f'(0)>0$ implies that there is a (small) $A>0$ such that $f(A)>0$ and $x<A \implies f(x)\le f(A)$.
$\lim_{x\rightarrow+\infty} f(x) = 0$ implies that there are (big) $B>A$ and (small) $b$ such that $f(A)>b>0$ such that $x>B \implies f(x)<b$.
Since $[A;B]$ is compact, $f$ reaches its maximum on it at $m\in[A;B]$ (by the Extreme value theorem), and the choice of $A$ and $B$ ensures that it is also a maximum on $[0;+\infty)$.
PPS. Without $f'(0)>0$ we cannot claim $m>0$ (only $m\ge0$).
I don't think this theorem is widely known under some specific name. Part of the reason might be that it is more or less an immediate consequence of the Extreme Value Theorem, as is seen by the proof you give. If I'd use "your" theorem, I would probably refer to it as just that.