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hw7-sol - Assignment 7 100 points total plus 10 bonus...

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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 ∇? = 𝜆∇? becomes ?? + ?? = 𝜆 8 ?? + 2 ?? . This gives 8 ?𝜆 = ? and 2 ?𝜆 = ? Multiplying, we get 8 ? 2 𝜆 = 2 ? 2 𝜆 If 𝜆 = 0 , then ? = ? = 0 , which does not satisfy the constraint equation. So 𝜆 ≠ 0 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 ?

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Page 2 ∇? = 2 ?? + ? + 2 ?? So, we have ∇? = 𝜆∇? ⇒ ? = 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 − ? 2 − ? 2 from the constraint function ? ? , ? , ? = 2 ? + ? + 4 ? = 2 ? + 16 − ?
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hw7-sol - Assignment 7 100 points total plus 10 bonus...

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