Optimization Now that we have discussed local maximums and minimums of functions, it is natural to consider applying these concepts to actual problems. In addition to just finding a local maximum or minimum, we might want to find the largest or smallest value on a region R. These values are called the global maximumand global minimumof the function on the region R. That is, we have Global Maximums and Global Minimums of Functions f(x, y) has a global maximum on a region R at the point (x0, y0) provided f(x0, y0) ¥f(x, y) for all points (x, y) in Rf(x, y) has a global minimum on a region R at the point (x0, y0) provided f(x0, y0) §f(x, y) for all points (x, y) in RA natural question is whether or not a function has a global maximum or global minimum value on a region R. Thinking back to one-variable calculus, the function y= x2has a global minimum at (0, 0), but goes off to infinity as the x-values get further away from 0. But if we specify a set of x-values, such as [-1, 2], then we can say that the function has a global maximum on [-1, 2]. In particular, the maximum occurs at x= 2. There was a result in one-variable calculus that showed that a continuous function (of one-variable) has a global maximum and a global minimum on [a, b] for constants aand b. An analogous result holds for functions of two or more variables. Extreme Value Theorem for Multivariable Functions If f(x, y) is a continuous function on a closed and bounded region R, then f(x, y) has a global maximum at some point (x1, y1) in Rand a global minimum at some points (x2, y2) in R. A region Ris said to be closed if the region contains is boundary. (In terms of a graph of the region, the region is outlined with solid lines as compared to dashed lines. This is analogous to requiring an interval [a, b] in one-variable calculus as opposed to (a, b), which did not include the end points.) A region is bounded if the values of the functions are finite (not infinity). 1
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