Let $a < b$, $]a,b[$ be an open real interval, $f:]a,b[\to\mathbb R$ be a twice differentiable function at a point $x\in ]a,b[$ with
$f'(x)=0$ and $f'\;'(x) > 0$ (respectively $f'\;'(x) < 0$).1
Then $x$ is a local maximum (respectively a local minimum) of $f$.
Proofs: 1
Note that the condition is only sufficient but not necessary for a local extremum. For instance, the function $f(x)=x^4$ has for $x=0$ a local minimum, but $f'\;'(x)=0.$ ↩
