optimization

optimization - Optimization Now that we have discussed...

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O ptimization 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 maximum and global minimum of 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 ( x 0 , y 0 ) provided f ( x 0 , y 0 ) ¥ f ( x , y ) for all points ( x , y ) in R f ( x , y ) has a global minimum on a region R at the point ( x 0 , y 0 ) provided f ( x 0 , y 0 ) § f ( x , y ) for all points ( x , y ) in R A 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 = x 2 has 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 a and 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 ( x 1 , y 1 ) in R and a global minimum at some points ( x 2 , y 2 ) in R . A region R is 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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Notice that the theorem does not tell us the locations of the points ( x 1 , y 1 ) and ( x 2 , y 2 ), but rather only that we can find them. The next natural question to ask is how we find those points. There are a lot (an infinite number) to choose from potentially. However, we only need to find critical points inside the region R and look for maximums and minimums (usually, but not always, critical points) on the boundary of R . The global maximum and the global minimum will occur from this collection of points. This should sound familiar to how one found global maximums and global minimums in one-variable calculus. As a reminder, here is the procedure we used before. How To Find Global Maximum and Global Minimum of a One- Variable Function on an interval [ a , b ] 1. Take the derivative of the function, set it equal to 0 and solve for x . These are critical points of the function.
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This note was uploaded on 09/07/2011 for the course MATH 10C taught by Professor Hohnhold during the Spring '07 term at UCSD.

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optimization - Optimization Now that we have discussed...

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