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

cp /math/calclab/MA1024/Optimization_start_C11.mw My_Documents

Another way to access the getting started worksheet is to go to your computer's Start menu and choose run. In the run field type:

\\filer\calclab

when you hit enter, you can then choose MA1024 and then choose the worksheet

Optimization_start_C11.mw.

Remember to immediately save it in your own home directory. Once you've copied and saved the worksheet, read through the background on the internet and the background of the worksheet before starting 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 by finding where both first partial derivatives are zero simultaneously, 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.

- Consider the function

Find the absolute extrema of this function over the region bounded by the rectangle , . - For the function

- Find and classify local extrema.
- Find and classify global extrema on the circlular region given by .

2011-02-01