14.7 - Section 15.7 Maximum and Minimum Values Finding...

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Section 15.7 Maximum and Minimum Values “Finding minimum and maximum values for functions of more than one variable” One important application in single variable calculus is ±nding the minimum and maximum values of a function. This can be done through the use of the ±rst and second derivative tests. In this section we shall develop tests to determine the minimum and maximum values of a function of two variables. 1. Minimum and Maximum Values First we need a formal de±nition of a minimum and maximum value. Defnition 1.1. A function of two variables f ( x, y ) has a local max- imum value at ( a, b ) if f ( x, y ) l f ( a, b ) whenever ( x, y ) is near ( a, b ). It is said to have a local minimum value at ( a, b ) if f ( x, y ) g f ( a, b ) whenever ( x, y ) is near ( a, b ). If the inequalities hold for all points ( x, y ) in the domain, we call ( a, b ) an absolute maximum or minimum of f ( x, y ). We call all such points extreme values. The following are examples of minimum and maximums. -1.0 -1.0 2.0 1.5 -0.5 x 1.0 -0.5 y 0.5 0.0 0.0 0.0 0.5 0.5 1.0 1.0 -1.25 -1.0 -0.75 -1.0 -1.0 -0.5 -0.5 -0.25 -0.5 x y 0.0 0.0 0.0 0.5 0.5 1.0 1.0 -3 -2 x -3 -1 -0.15 y -2 -0.1 -1 -0.05 0 0 0.0 0.05 1 0.1 2 1 0.15 3 2 3 The following result will help us to ±nd minimum and maximum values for f ( x, y ). Result 1.2. If f has a local maximum or minimum at ( a, b ) and the ±rst order partial derivatives exist, then f y ( a, b ) = f x ( a, b ) = 0. This is a direct generalization of the single variable case. We make a couple of important observations: ( i ) Just because the partial derivatives are 0 does not mean there is a min or max (we shall look at some examples of this later). ( ii
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This note was uploaded on 10/28/2010 for the course MA 261 taught by Professor Stefanov during the Fall '08 term at Purdue University-West Lafayette.

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14.7 - Section 15.7 Maximum and Minimum Values Finding...

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