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10 Notes 8 - MATH 20C Lecture 19 Monday November 8 2010...

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MATH 20C Lecture 19 - Monday, November 8, 2010 Lagrange multipliers continued Example: Find the min/max of f ( x, y, z ) = 3 x + y + 4 z on the surface x 2 + 3 y 2 + 6 z 2 = 1 . Step 1 Compute the two gradients f and g. f = h 3 , 1 , 4 i g = h 2 x, 6 y, 12 z i Step 2 Write down the Lagrange multiplier equations f = λ g and the constraint g = c. f = λ g = 3 = 2 λx 1 = 6 λy 4 = 12 λz g = c = x 2 + 3 y 2 + 6 z 2 = 1 Step 3 Solve the system, i.e. find points ( x, y, z ) that satisfy the equations from Step 2. WARNING! There is no general method to solve these equations. In each case, you have to think about them and come up with a method. Sometimes it will be impossible to solve without using a computer. (Not on the exam though!) In our example, note that λ cannot be 0 . From the first three equations get x = 3 2 λ , y = 1 6 λ , z = 1 3 λ . Substitute these values in the constraint equation and get λ 2 = 3 , so λ = ± 3 . Therefore there are two points 3 2 , 1 6 3 , 1 3 3 and - 3 2 , - 1 6 3 , - 1 3 3
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