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Assignment 5

# Assignment 5 - = x 2 y 2-1 2 y a Use the method of Lagrange...

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Math 237 Assignment 5 Due: Friday, Oct 31st 1. Find and classify the critical points of the following functions and determine the shape of the level curves near each critical point. a) f ( x, y ) = 1 2 x 2 + 3 y 3 + 9 y 2 - 3 xy + 9 y - 9 x . b) g ( x, y ) = x 3 + y 3 - 6 y 2 - 3 x + 9. c) h ( x, y ) = ( x + y )( xy + 1). 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. 3. Find the maximum and minimum of f ( x, y ) = x 2 - y 2 on the region x 2 + y 2 1. 4. Find the points on the surface z = x 2 + y 2 that is closest to the point (1 , 1 , 0). 5. Let
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Unformatted text preview: ) = x 2 + y 2-1 2 y . a) Use the method of Lagrange multipliers to ﬁnd the maximum and minimum points of f ( x,y ) on the curve y = √ 1-2 x 2 . b) Let R be the region bounded by the curve y = √ 1-2 x 2 and the x-axis. Find the maximum and minimum value of f ( x,y ) on the region R . 6. Find the maximum and minimum values of f ( x,y ) = y on the piriform curve deﬁned by y 2 + x 4-x 3 = 0....
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