53 Worksheet 10_10 Solutions

53 Worksheet 10_10 Solutions - Math 53: Multivariable...

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Math 53: Multivariable Calculus Worksheet 10/9/11: Lagrange Multipliers Exercise 0.1. Maximize f ( x,y ) = x 2 y , subject to the constraint that x 2 + 2 y 2 = 6. Solution. Set g ( x,y ) = x 2 + 2 y 2 . Setting 5 f = λ 5 g and g ( x,y ) = 6, we have 2 xy = λ 2 x x 2 = λ 4 y x 2 + 2 y 2 = 6 The ﬁrst equation gives 2 x ( y - λ ) = 0, so either x = 0 or λ = y . If x = 0, we get 0 2 + 2 y 2 = 6, so y = ± 3, and our points are (0 , 3) and (0 , - 3). If λ = y , we have x 2 = 4 y 2 from the second equation. Plugging this into the third equation gives 4 y 2 + 2 y 4 = 6, so y 2 = 1 and y = ± 1. This means x 2 + 2 = 6, so x = ± 2. This gives the points (2 , 1), ( - 2 , 1), (2 , - 1), and ( - 2 , - 1). Plugging in our candidate points gives f (0 , 3) = 3 f (0 , - 3) = - 3 f (2 , 1) = 4 f (2 , - 1) = - 4 f ( - 2 , 1) = 4 f ( - 2 , - 1) = - 4 So the min is - 4 (attained at (2 , - 1) and ( - 2 , - 1)) and the max is 4 (attained at (2 , 1) and ( - 2 , 1)). ± Exercise 0.2.

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53 Worksheet 10_10 Solutions - Math 53: Multivariable...

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