** Next:** About this document ...
**Up:** No Title
** Previous:** No Title

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.

A crucial first step in solving such problems is being able to find and classify local extreme points of a function. What we mean by the term local extreme values is contained in the following definition.

**Definition 188**

Let *f* be a function defined at a point (*x _{0}*,

In single-variable calculus, we found that the first derivative vanished at a local extreme value. For functions of two variables, both first-order partial derivatives vanish as described by the following theorem.

**Theorem 194**

If a function *f* has a local extreme value at a point (*x _{0}*,

Notice that having both first order partial derivatives vanish means
that the tangent plane is horizontal.
Following the terminology we used for functions of a single variable,
we call points where the partial derivatives *f*_{x} and *f*_{y} vanish
*critical points*.
Note carefully that the theorem does not say that a point where the partial
derivatives vanish must be a local extreme point. Rather, it says that
critical points are candidates for local extrema. Just as was the case
for functions of a single variable, there can be critical points that
are not extrema. For example, the saddle surface *f*(*x*,*y*) = *x ^{2}*-

Finding and classifying the local extreme values of a function
*f*(*x*,*y*) requires several steps. First, the partial derivatives must
be computed. Then the critical points must be solved for, which is not
always a simple task. 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.

**Theorem 209**

Suppose *f*(*x*,*y*) is continuous on a region *S* bounded by a simple closed
curve, including the boundary. Then *f*(*x*,*y*) attains its absolute
maximum value at some point (*x _{0}*,

This theorem only says that the extrema exist, but doesn't help at all
in finding them. However, we know that the global extrema occur either
at local extrema, on the boundary of the region, or at points where
one or the other partial derivative fails to exist. For example, to find the extreme values of a
function *f*(*x*,*y*) on the rectangle given above, you would first have to
find the interior critical points and then find the extreme values for
the four one-dimensional functions

- 1.
- Find and classify the critical points for the following
functions.
- (a)
*f*(*x*,*y*) =*x*+2*xy*+*y*.^{2}- (b)
- .
- (c)
- .(Hint - there are 4.)

- 2.
- Consider the function
Find the absolute extrema of this function on the following domains.
- (a)
- The rectangle , .
- (b)
- The rectangle , .
- (c)
- The set of points in the plane satisfying .

- 3.
- What is the maximum possible volume of a rectangular box
inscribed in a hemisphere of radius
*R*? You may assume that one face of the box lies in the planar base of the hemisphere.

4/11/2000