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Homework4Solns - SHORT CALCULUS Math 16C Sec 2 Spring 2008...

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SHORT CALCULUS Math 16C Sec 2 Spring 2008 Homework #4 Solutions Peter Malkin Section 7.6 Question 2 Let F ( x, y, λ ) = xy - λ (2 x + y - 4). F x = y - 2 λ = 0 λ = 1 2 y (1) F y = x - λ = 0 λ = x (2) F λ = - (2 x + y - 4) = 0 (3) Using equation (1) and (2) gives 1 2 y = x y = 2 x . Putting this into equation (3) gives 2 x + 2 x - 4 = 0 x = 4 , so y = 2. Thus, f ( x, y ) = xy is maximum at the point (1 , 2), and the maximum value is f (1 , 2) = 2. Question 6 Let F ( x, y, λ ) = x 2 - y 2 - λ ( x - 2 y + 6). F x = 2 x - λ = 0 (4) F y = - 2 y + 2 λ = 0 (5) F λ = - ( x - 2 y + 6) = 0 (6) Equation (1) implies λ = 2 x , and substituting this into (2) gives, - 2 y + 4 x = 0, which implies y = 2 x . Substituting this into (3) gives - ( x - 4 x + 6) = 0 x = 2 . So, y = 2 x = 4. Thus, f ( x, y ) is maximum at (2 , 4), and the maximum value is f (2 , 4) = - 12. Question 16 Let F ( x, y, λ ) = x 2 - 8 x + y 2 - 12 y + 48 - λ ( x + y - 8). F x = 2 x - 8 - λ = 0 (1) F y = 2 y - 12 - λ = 0 (2) F λ = - ( x + y - 8) = 0 (3) (1) - (2) : 2 x - 2 y + 4 = 0 x - y + 2 = 0 (4) (4) + (3) : - 2 y + 10 = 0 y = 5 (5) Substituting y = 5 into (3) gives - ( x +5 - 8) = 0 x = 3. So, f ( x, y ) is maximum at (3 , 5), and the maximum value is f (3 , 5) = 3 2 - 8 · 3 + 5 2 - 12 · 5 + 48 = - 2. 1
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Question 18 Let F ( x, y, z, λ ) = x 2 y 2 z 2 - λ ( x 2 - y 2 - z 2 - 1). F x = 2 xy 2 z 2 - λ 2 x = 0 2 x ( λ - y 2 z 2 ) = 0 x = 0 or λ = y 2 z 2 (1) F y = 2 yx 2 z 2 - λ 2 y = 0 2 y ( λ - x 2 z 2 ) = 0 y = 0 or λ = x 2 z 2 (2) F z = 2 zx 2 y 2 - λ 2 z = 0 2 z ( λ - x 2 y 2 ) = 0 z = 0 or λ = x 2 y 2 (3) F λ = - ( x 2 - y 2 - z 2 - 1) = 0 (4) Note that we can ignore the cases where x = 0, y = 0, or z = 0 since they do not give a maximum. Then, (1) and (2) imply y 2 z 2 = λ = x 2 z 2 z = 0 or y 2 = x 2 x = y (since x > 0 y > 0). Then, (1) and (3) imply y 2 z 2 = λ = x 2 y 2 y = 0 or z 2 = x 2 z = x (since x > 0 z > 0). Again, we can ignore when z = 0 or y = 0. Then, substituting y = x and z = x
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