Extrema: maximum and minimum

Let's see now the definitions of relative and absolute extrema by means of an example:

• A function f has a relative or local minimum (respectively relative or local maximum) in $$x = a$$, if there is a neighbourhood $$E_r(a)$$ of the point, such that for every $$x$$ belonging to $$E_r(a)$$, we have $$$f(x) \leq f(a) (\mbox{ respectively }f(x) \geq f(a))$$$

• A function $$f$$ has an absolute minimum (respectively absolute maximum) in $$x = a$$, if for any $$x$$ of the domain of $$f$$ we have $$$f(x) \leq f(a) (\mbox{ respectively }f(x) \geq f(a))$$$