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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 finding the minimum and maximum values of a function. This can be done through the use of the first 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 definition of a minimum and maximum value. Definition 1.1. A function of two variables f ( x,y ) has a local max- imum value at ( a,b ) if f ( x,y ) lessorequalslant 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 ) greaterorequalslant 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 find minimum and maximum values for f ( x,y ). Result 1.2. If f has a local maximum or minimum at ( a,b ) and the first 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).

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

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