Given a function $$f$$, a point $$\xo$$ is
- a local maximum if, for all $$x$$ around $$\xo$$, $$$f(x)\le f(\xo);$$$
- a local minimum if, for all $$x$$ around $$\xo$$, $$$f(x)\ge f(\xo).$$$
An extremum is either a minimum or a maximum. A local extremum is sometimes called a turning point or a relative extremum.
$$0$$ is a minimum of $$ x^2$$ and a maximum of $$ -x^2$$.
At a local extremum, the tangent must be horizontal $$$ f'(\xo)=0. $$$
At a local maximum $$\xo$$, $$f$$ "increases" to its left, so $$f'(\xo)\ge 0$$, and "decreases" to its right, so $$f'(\xo)\le 0$$. Thus, $$f'(\xo)=0$$.