Extrema

The upper and lower bounds of a real valued function are of interest in several situations in pure as well as applied Mathematics

Absolute Extrema

Let $A$ be a set

Let $f:A\rightarrow\mathbb{R}$

Let the set $f(A)$ be bounded

Then $M=\sup\{f(A)\}$ is called the Absolute or Global maximum of $f$

and $m=\inf\{f(A)\}$ is called the Absolute or Global minimum of $f$

Let $f:[a,b]\rightarrow\mathbb{R}$

Let $c\in [a,b]$

$f(c)$ is said to be a Local maximum iff $\exists\delta>0$ such that $f(c)=\sup\{f(V_{\delta}(c))\}$

$f(c)$ is said to be a Local minimum iff $\exists\delta>0$ such that $f(c)=\inf\{f(V_{\delta}(c))\}$