a7_soln

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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online