MATH23 Homework 7 Solution - Homework#7 Solutions Math 23...

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Homework #7 Solutions Math 23, Spring 2015 1. Find the maximum value of f ( x, y, z ) = 6 x + 8 y - 2 z on the sphere x 2 + y 2 + z 2 = 26 . We use Lagrange multipliers. We see that the sphere is the k = 26 level surface of g ( x, y, z ) = x 2 + y 2 + z 2 . We compute f = h 6 , 8 , - 2 i and g = h 2 x, 2 y, 2 z i . So we need to solve the system of equations 6 = 2 λx 8 = 2 λy - 2 = 2 λz x 2 + y 2 + z 2 = 26 . In each of the first three equations, we can solve for x , y or z in terms of λ . (Note that we can’t have λ = 0, since then the first equation, for example, would have no solution.) This gives x = 3 , y = 4 , and z = - 1 . Using these in the last equation gives 9 λ 2 + 16 λ 2 + 1 λ 2 = 26 which simplifies to 26 λ 2 = 26 . So we have that λ = ± 1. Substituting these two values into the equations for x , y , and z , we find two critical points, (3 , 4 , - 1) and ( - 3 , - 4 , 1). Evaluating f at these points, we find f (3 , 4 , - 1) = 52 and f ( - 3 , - 4 , 1) = - 52. Thus, the maximum value of f on this sphere is 52.