4. Finding Extreme Values

Critical Points

Definition

A point $p$ in the domain of a function $f$ of $n$ variables is called a critical point of $f$ if

• $f_{x_i}(p)=0$ for $i=1,\dots ,n$ or
• one of the partial derivatives fails to exist at $p$.

Theorem

Suppose $f(x)$ has a local minimum or local maximum at the point $p$. Then $p$ is a critical point of $f$.

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.

Corollary

In the setting of the previous theorem the function must attain its maximum and minimum either on the boundary of the domain or at a critical point of interior of the domain.

Closed and Bounded Region Method

Algorithm

To find the absolute maximum and minimum values of a continuous function $f$ on a closed and bounded domain $D$ apply the following algorithm:

1. Find the values of $f$ at the critical points of $f$ in the interior of the region $D$.
2. Find the extreme values of $f$ restricted to the boundary of the region $D$ by applying this algorithm to the function obtained by restricting $f$ to the boundary of $D$.
3. The largest of the values from Steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.