Local extremum : A local extremum of a function $f$ is an $x_0$ such that there exist $a,b$ with $a\lt x_0\lt b$ and $f(x_0)\geq f(x)$ for all $a\lt x\lt b$ or $f(x_0)\leq f(x)$ for all $a\lt x\lt b.$ $ A local extremum can but does not have to be a global extremum.