Find the maximum value of 2x + 2y + z on the sphere of radius 1 at the origin.

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- Nov 15th 2010, 05:07 PMPlaythiousLagrange Multipliers
Find the maximum value of 2x + 2y + z on the sphere of radius 1 at the origin.

- Nov 15th 2010, 05:11 PMdwsmith
So your constraint is

Now find:

Then set the i, j, and k components equal to each other of f and g where f is the function and g is the constraint. - Nov 15th 2010, 05:31 PMPlaythious
I've found lambda(X) = 1

lambda(Y) = 1

lambda(Z) = 1/2

and x^2 + y^2 + z^2 = 1

how would I solve this for a maximum, taking into account the case when lambda is 0? - Nov 15th 2010, 05:39 PMdwsmith
I am just typing this up so I can see it.

Since we want the max, we take the positive value.

Now plug into f. - Nov 15th 2010, 06:12 PMPlaythious
ok

thanks for your help - Nov 16th 2010, 08:00 AMHallsofIvy
Since the value of is not part of the solution, you can often simplify by first eliminating by

**dividing**one equation by another.

Here, you have and so, dividing the first by the second,

so that

or y= x. Similarly, dividing by gives x= 2z.

Putting those into gives .