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# Class Note for MATH 1314 at UH

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## About this Document

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This 5 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 15 views.

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Date Created: 02/06/15
Math 13 14 Maxima and Minima of Functions of Several Variables Relative Extrema of Functions of Two Variables We learned to nd the maxima and minima of a function of a single variable earlier in the course Although we did not use it much we had a second derivative test to determine whether a critical point of a function was a maximum or a minimum or possibly that the test was not conclusive We will use a similar technique to find relative extrema of a function of several variables Since the graphs of these functions are more complicated determining relative extrema is also more complicated At a specific critical number we can have a max a min or something else That something else is called a saddle point wail 3 355334 W I 390quot v 3 15 a if a 4 49 v 333 o s r I gag v The method for finding relative extrema is very similar to what you did earlier in the course First nd the first order partial derivatives and set them equal to zero You will have a system of equations in two variables which you will need to solve to find the critical points Second you will apply the second derivative test To do this you must find the second order partial derivatives Let Dx y fmfyy ny 2 You will compute Da b for each critical point a b Then you can apply the second derivative test for functions of two variables 0 IfDa b gt 0 and f a b lt 0 thenfhas a relative maximum at a b 0 IfDa b gt 0 and f a b gt 0 thenfhas a relative minimum at a b 0 If Da b lt 0 then fhas neither a relative maximum nor a relative minimum at a b ie it has a saddle point which is neither a max nor a min 0 IfDa b 0 then this test is inconclusive Example 1 Find the relative extrema of the function f x y x2 yz Example 2 Find the relative extrema ofthe function fx y x3 y2 6x2 8y 15x 7 Example 3 Find the relative extrema ofthe function fx y x3 2y2 2xy 3x 2y 5 Example 4 Find the relative extrema of the function f x y 3x3 4xy 2 y2 7 Example 5 The total daily revenue in dollars that a publishing company realizes in publishing and selling its English language dictionary is given by Rx y 005x2 003y2 002xy 20x 17y where x denotes the number of deluxe copies and y denotes the number of standard copies published and sold The total daily cost of publishing these dictionaries is given by C x y 6x 3y 200 dollars Determine the number of standard copies and the number of deluxe copies that the publishing company should publish per day to maximize its pro ts What is the maximum pro t realizable From this section you should be able to Find relative extrema of lnctions of two variables Solve word problems involving extrema of functions of two variables

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