hw7-sol - Page 1 Assignment 7 100 points total plus 10...

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Unformatted text preview: Page 1 Assignment 7 100 points total plus 10 bonus points Section 15.3 Problem 6 : Our objective function is ? ? , ? = ?? and our equation of constraint is ? ? , ? = 4 ? 2 + ? 2 = 8 . Their gradients are ? ? , ? = ? + ? ? ? , ? = 8 ? + 2 ? So the equation ? = ? becomes ? + ? = 8 ? + 2 ? . This gives 8 ? = ? and 2 ? = ? Multiplying, we get 8 ? 2 = 2 ? 2 If = 0 , then ? = ? = 0 , which does not satisfy the constraint equation. So and we get 2 ? 2 = 8 ? 2 ? = 2 ? To find ? , we substitute for ? in our equation of constraint. 4 ? 2 + ? 2 = 8 4 ? 2 + 4 ? 2 = 8 ? = 1 So our critical points are 1,2 , 1, 2 , ( 1,2) and 1, 2 . Since the constraint is closed and bounded, maximum and minimum values of ? subject to the constraint exist. Evaluating ? ? , ? at the critical points, we have ? 1,2 = ? 1, 2 = 2 ? 1, 2 = ? 1,2 = 2 Thus, the maximum value of ? on ? ? , ? = 8 is 2, and the minimum value is 2 . Problem 10 : The gradients of the objective and the constraint functions are ? = 2 + + 4 ? Page 2 ? = 2 ? + ? + 2 ? So, we have ? = ? ? = 1, ? = 2 Going back to the constraint function, we can solve for = 11 . This gives us one critical point 1,11,2 . ? 1,11,2 = 21 This is the maximum value of ?? , , ? on ?? , , ? = 16 . To see this, we note that = 16 ?...
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This note was uploaded on 03/21/2010 for the course AMS 261 taught by Professor Fortmann during the Fall '08 term at SUNY Stony Brook.

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hw7-sol - Page 1 Assignment 7 100 points total plus 10...

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