Subsections

• Purpose
• Background
  • Locating Global Extrema
• 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.

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 points of a function. What we mean by the term local extreme values is contained in the following definition.

Definition 1

Let f be a function defined at a point (x₀,y₀). Then f(x₀,y₀) is a local maximum if $f(x_0,y_0) \geq f(x,y)$ for all (x,y) in an open disk containing ( x₀,y₀) and f(x₀,y₀) is a local minimum if $f(x_0,y_0) \leq f(x,y)$ for all (x,y) in an open disk containing ( x₀,y₀). If f(x₀,y₀) is a local maximum or a local minimum, we say that it is a local extreme value.

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 1

If a function f has a local extreme value at a point (x₀,y₀) and the partial derivatives of f both exist at (x₀,y₀), then

$$\frac{\partial f}{\partial x}(x_0,y_0) = \frac{\partial f}{\partial y}(x_0,y_0) = 0$$

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) = x²-y² 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.

Locating Global Extrema

In one-dimensional calculus, the absolute or global extreme values of a function occur either at a point where the derivative is zero, a boundary point, or where the derivative fails to exist. The situation for a function of two variables is very similar, but the problem is much more difficult because the boundary now consists of curves instead of just endpoints of intervals. For example, suppose that we wanted to find the global extreme values of a function f(x,y) on the rectangle $S = \{(x,y):\, a \leq x \leq b \mbox{ and } c \leq y \leq d\}$. The boundary of this rectangle consists of the four line segments given below.

$$\begin{array} {lcrc} y & = & c, & a \leq x \leq b \\ y & = &... ...a, & c \leq y \leq d \\ x & = & b, & c \leq y \leq d\end{array}$$

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

Theorem 2

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 (x₀,y₀) in S and absolute minimum value at some point x₁,y₁) in S.

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

$$\begin{array} {lccc} g_1(x) & = & f(x,c), & a \leq x \leq b ... ...q y \leq d \\ h_2(y) & = & f(b,y), & c \leq y \leq d\end{array}$$

Exercises

1.

Find and classify the critical points for the following functions.

(a)

f(x,y) = x²+2xy+y.

(b)

$f(x,y) = (x^2-2x)\sin(y)$.

(c)

$f(x,y) = (x^2+y)\exp(-x^2-y^2)$.(Hint - there are 4.)

2.

Consider the function

$$f(x,y) = \frac{x+y}{1+x^2+y^2}$$

Find the absolute extrema of this function on the following domains.

(a)

The rectangle $-2 \leq x \leq 2$, $-2 \leq y \leq 2$.

(b)

The rectangle $-2 \leq x \leq 0$, $-2 \leq y \leq 0$.

(c)

The set of points in the plane satisfying $x^2+y^2 \leq 1/2$.

3.

A rectangular box is inscribed in the tetrahedron formed in the first octant by the coordinate planes and the plane x/a+y/b+z/c=1 where a, b, c > 0. Assume that the faces of the box are parallel to the coordinate planes. What are the dimensions of the box of largest volume?