432hwk3 - x 2 y 2 z 2 )-x-y-z (e) f ( x, y, z ) = x 2 + y 2...

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Math 432/532 Homework 20 pts. Due Feb. 3, 2012 1. For the following functions, all critical points are listed. Identify all local and global maxima and minima, along with saddle points. (a) f ( x, y ) = x 3 + y 3 - 3 x - 12 y + 20 Critical points: (1 , 2) , (1 , - 2) , ( - 1 , 2) , ( - 1 , - 2) (b) f ( x, y ) = x 4 + y 4 - x 2 - y 2 + 1 Critical points: (0 , 0) , (0 , ± 1 / 2) , ( ± 1 / 2 , 0) , ( ± 1 / 2 , 1 / 2) , ( ± 1 / 2 , - 1 / 2) (c) f ( x, y ) = 12 x 3 - 36 xy - 2 y 3 + 9 y 2 - 72 x + 60 y + 5 Critical points: (0 , - 2) , (1 , - 1) , (2 , 2) , ( - 3 , 7) (d) f ( x, y ) = e - ( x 2 + y 2 ) Critical point: (0 , 0) (e) f ( x, y ) = x 4 + 16 xy + y 8 Critical points: (0 , 0) , (2 3 / 4 , - 2 1 / 4 ) , ( - 2 3 / 4 , 2 1 / 4 ) 2. Which of the following functions are coercive? For those that are not coercive, explain why they are not. (a) f ( x, y, z ) = x 3 + y 3 + z 3 - xy (b) f ( x, y, z ) = x 4 + y 4 + z 2 - 3 xy - z (c) f ( x, y, z ) = x 4 + y 2 + z 4 - 5 xy 2 z (d) f ( x, y, z ) = ln(
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Unformatted text preview: x 2 y 2 z 2 )-x-y-z (e) f ( x, y, z ) = x 2 + y 2 + z 2-cos( x 2 + y 2 + z 2 ) 3. (532 only) Suppose that f : R n R is a function with continuous second-order partial derivatives whose Hessian is positive denite at all points in R n . (a) Show that f has at most one critical point. (b) Show, by example, that f may have no critical points. (c) Show that if x * is a critical point of f , then x * is a strict global minimizer of f . (d) Show that if f ( x (1) ) = f ( x (2) ), then x (1) = x (2) ....
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