2. Extreme Values

Absolute Extreme Values

Definition

Let $p=(p_1,\dots ,p_n)$ be a point in the domain $D$ of a function $f(x)$. Then $f(p)$ is the

• Absolute Maximum value of $f$ on $D$ if $f(p)\ge f(x)$ for all $x$ in $D$.
• Absolute Minimum value of $f$ on $D$ if $f(p)\le f(x)$ for all $x$ in $D$.

Local Extreme Values

Definition

Let $p=(p_1,\dots ,p_n)$ be a point in the domain $D$ of a function $f(x)$. Then $f(p)$ is a

• Local Maximum value of $f$ if there exists a number $R>0$ such that $f(p)\ge f(x)$ for all $x$ such that $||x-p||<R$.
• Local Minimum value of $f$ if there exists a number $R>0$ such that $f(p)\le f(x)$ for all $x$ such that $||x-p||<R$.

Extreme Value Theorem

Theorem

Suppose $f(x)$ is a continuous function on a closed and bounded domain $D\subset R^n$. Then $f$ attains both an absolute maximum and an absolute minimum on that domain.

Bounded Domains

Definition

A region $D\subset R^n$ is called bounded if there exist a number $M>0$ such that $||x||<M$ for all $x\in D$.

Closed Domain

Definition

A region $D\subset R^n$ is called closed if it contains all of its boundary points.