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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.
Let f be a function defined at a point (x0,y0). Then f(x0,y0) is a local maximum if for all (x,y) in an open disk containing ( x0,y0) and f(x0,y0) is a local minimum if 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.
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
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 fx and fy 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.
The basic theorem on the existence of global maximum and minimum values is the following.
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
Find the absolute extrema of this function on the following domains.
William W. Farr