Subsections

• Purpose
• Getting Started
• Background
• Exercises

Finding and Classifying Extreme Values

Purpose

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.

Getting Started

To assist you, there is a worksheet associated with this lab that contains examples. You can copy that worksheet to your home directory with the following command, which must be run in a terminal window, not in Maple.

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.

Background

Many applications of calculus involve finding the maximum and minimum values of functions. For example, suppose that there is a network of electrical power generating stations, each with its own cost for producing power, with the cost per unit of power at each station changing with the amount of power it generates. An important problem for the network operators is to determine how much power each station should generate to minimize the total cost of generating a given amount of power.

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 $f$ having a local extreme value at a point $\mathbf{p}_0$ is that for values of $\mathbf{p}$ near $\mathbf{p}_0$, $f(\mathbf{p}_0) \geq f(\mathbf{p})$ for a local maximum and $f(\mathbf{p}_0) \leq f(\mathbf{p})$ 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 $f(x,y) = x^2-y^2$ has a critical point at the origin, but it is not a local extremum.

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.

Exercises

1. Find and classify the stationary points for the following functions.
   1. $$f(x,y) = x^2+4y^2-2x+8y-1$$

   2. $$r(x,y) =\frac{y-x}{(4+x^2)(4+y^2)}$$

2. A rectangular shipping crate is to have volume $12 {\mathrm ft}^3$. The material for the bottom of the crate costs $2 per square foot and the material for the top and sides costs $1 per square foot. What dimensions of the crate will minimize the total cost?