PS5_answers

PS5_answers - b) An example of a 4-saddle is f ( x,y ) = xy...

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Math 237 Problem Set 5 Answers A1. i) (0 , 1), (1 , 1) and (0 , 0) are all saddle points. ( 1 3 , 2 3 ) is a local min. ii) (0 , 0) is a saddle point, ( - 1 , - 1 / 2) is a local max. iii) (0 , 1 / 3) local min, (0 , - 1 / 3) local max, ( ± 1 , 0) are saddle points. iv) (0 ,kπ ), k Z are all saddle points. A2. The min is 0 along x = 0, 0 y 1 and y = 0, 0 x 1. The max is 2 3 3 at ( 1 3 , 1). A3. Max 2 5, Min - 2 5. A4. Max 3 2 e at (1 , 3 / 2) and min is 0 at x = 0, 0 y 3 and y = 0, 0 x 2. A5. Hottest at ± - b 2 , 3 b 2 ² , ± - b 2 , - 3 b 2 ² and coldest at ± b 2 , 3 b 2 ² , ± b 2 , - 3 b 2 ² . A6. a) Greatest distance 7, least distance 2. A7. The closest distance is q 125 4 . A8. Max 14. We have found the point of intersection of the plane x + y + z = 14 and the ellipsoid where the gradient vectors are parallel. A10. Max 7, Min 1 B1. Let x be the length of the angled sides and let θ be the angle the sides make with the horizontal. Then the maximum possible flow is when x = L 3 and θ = π 3 . B3. Max 5, Min 1 B4. a) (0 , 0) is the only critical point and it is a saddle point.
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Unformatted text preview: b) An example of a 4-saddle is f ( x,y ) = xy ( y 2-x 2 ); an n-saddle is g ( x,y ) = ( y-x )( y-1 2 x )( y-1 3 x ) ( y-1 n x ). B5. a) (0 , 0) is a saddle point so the level curves near (0 , 0) are hyperbola. ( b, b ) are both local minima and so the level curves near both these points are ellipses. B6. k 1, (0 , 0) is a global max, no local or global min. 0 < k < 1 (0 , 0) is a local min all pointson x 2 + y 2 = 1-k are global max. No global min. k < 0, (0 , 0) is a global min, all points x 2 + y 2 = 1-k are global max. B7. i) f ( x,y ) = 2( x-1 2 y ) 2 . The critical points y = 2 x are all local minimums. ii) f ( x,y ) = p | 1-x 2-y 2 | . The critical points x 2 + y 2 = 1 are all local minimums, (0 , 0) is a local maximum. 1...
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This note was uploaded on 04/18/2010 for the course MATH 237 taught by Professor Wolczuk during the Spring '08 term at Waterloo.

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