10 Notes 8

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

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Unformatted text preview: 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 = g and the constraint g = c. f = g = 3 = 2 x 1 = 6 y 4 = 12 z 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 cannot be 0 . From the first three equations get x = 3 2 ,y = 1 6 ,z = 1 3 . Substitute these values in the constraint equation and get 2 = 3 , so = 3 ....
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This note was uploaded on 04/28/2011 for the course MATH 20C taught by Professor Helton during the Spring '08 term at UCSD.

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

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