a7_soln - Math 237 Assignment 7 Solutions 1 Find and...

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Math 237 Assignment 7 Solutions 1. Find and classify the critical points of f ( x, y ) = ( x + y )( xy + 1). Solution: We need to solve 0 = f x = xy + 1 + y ( x + y ) = 2 xy + y 2 + 1 0 = f y = xy + 1 + x ( x + y ) = 2 xy + x 2 + 1 Combining these we get y 2 + 1 = - 2 xy = x 2 + 1 and hence x 2 = y 2 . If x = y then we have 0 = 3 y 2 + 1 which has no solutions. If x = - y then we have - y 2 + 1 = 0 and so y = ± 1 which gives points (1 , - 1) and ( - 1 , 1). We have Hf ( x, y ) = ± 2 y 2 x + 2 y 2 y + 2 x 2 x ² . At (1 , - 1) we get Hf (1 , - 1) = ± - 2 0 0 2 ² so det Hf (1 , - 1) = - 4 < 0 so it is indefinite and hence (1 , - 1) is a saddle point. At ( - 1 , 1) we get Hf ( - 1 , 1) = ± 2 0 0 - 2 ² so det Hh ( - 1 , 1) = - 4 < 0 so it is indefinite and hence ( - 1 , 1) is a saddle point. 2. Find the maximum and minimum of f ( x, y ) = x 3 - 3 x + y 2 + 2 y on the region bounded by the lines x = 0, y = 0, x + y = 1. Solution: We first look for critical points inside the region. We have 0 = f x = 3 x 2 - 3 = 3( x - 1)( x + 1) 0 = f y = 2 y + 2 = 2( y + 1) Thus, the critical points are (1 , - 1) and ( - 1 , - 1) neither of which are in the region. We then check the boundary of the region. For the line
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.

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a7_soln - Math 237 Assignment 7 Solutions 1 Find and...

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