Real function/Local extremum/Differentiable/Derivative zero/Fact

Let ${}D\subseteq \mathbb {R}$ denote an open subset and let

$f\colon D\longrightarrow \mathbb {R}$

be a function, which attains in ${}a\in D$ a local extremum and is differentiable there.

Then

${}f'(a)=0\,$

holds.

Proof, Write another proof