The purpose of this lab is to acquaint you with techniques for finding and classifying local and global extreme values of functions of two variables.

cp ~bfarr/Extrema2D_start.mws ~

You can copy the worksheet now, but you should read through the lab
before you load it into Maple. Once you have read to the exercises,
start up Maple, load
the worksheet `Extrema2D_start.mws`, and go through it
carefully. Then you can start working on the exercises.

A crucial first step in solving such problems is being able to find and classify local extreme values of a function. What we mean by a function having a local extreme value at a point is that for values of near , for a local maximum and for a local minimum.

In single-variable calculus, we found that we could locate candidates for local extreme values by finding points where the first derivative vanishes. For functions of two dimensions, the condition is that both first order partial derivatives must vanish at a local extreme value candidate point. Such a point is called a stationary point. It is also one of the three types of points called critical points. Note carefully that the condition does not say that a point where the partial derivatives vanish must be a local extreme point. Rather, it says that stationary points are candidates for local extrema. Just as was the case for functions of a single variable, there can be stationary points that are not extrema. For example, the saddle surface has a stationary point at the origin, but it is not a local extremum.

Finding and classifying the local extreme values of a function
requires several steps. First, the partial derivatives must
be computed. Then the stationary points must be solved for, which is not
always a simple task. Next, one must check for the presence of
*singular points*, which might also be local extreme
values. Finally, each
critical point must be classified
as a local maximum, local minimum, or neither. The examples in the
`Getting Started` worksheet
are intended to help you learn how to use Maple to simplify these tasks.

The basic theorem on the existence of global maximum and minimum values is the following.

- Find and classify the stationary points for the following
function.

- Consider the function

Find the absolute extrema of this function on the rectangle , . - What is the maximum possible volume of a rectangular box inscribed in a hemisphere of radius ? You may assume that one face of the box lies in the planar base of the hemisphere.

2004-02-13