Find the maximum and minimum values of f(x,y) = 2x^2+4y^2 - 4xy -4x on the circle defined by x^2+y^2 = 16.

Began using Lagrange's method but could not solve the resulting system. Any help is much appreciated.

Question

Find the maximum and minimum values of f(x,y) = 2x^2+4y^2 - 4xy -4x on the circle defined by x^2+y^2 = 16. 

Began using Lagrange's method but could not solve the resulting system. Any help is much appreciated.

jamesj · Answer

Writing the constraint equation as
g(x,y) = x^2 + y^2, we have, differentiating wrt x:

4x - 4y - 4 - L.2x = 0

wrt y:

8y - 4x - L.2y = 0

where L is lambda.

Agreed?

anonymous · Answer

That's right. Can't seem to solve that system along with the original constraint.

jamesj · Answer

I haven't worked this problem, but as a brute force method, I would begin by using both of these equations to write down expressions for Lambda.  Then set them equal to each other. 

Then see what you get and try and combine it with the circle x^2 + y^2 = 16.

anonymous · Answer

Gave that a try and ran into trouble but will try again. Thanks for the help.

jamesj · Answer

These are messy. This might help you: http://www.wolframalpha.com/input/?i=min+f%28x%2Cy%29+%3D+2x%5E2%2B4y%5E2+-+4xy+-4%2C+x%5E2+%2B+y%5E2+%3D+16

anonymous · Answer

Yeah, I feel there may be a typo in the question.

jamesj · Answer

Yes, when I do what I suggested, I find that
$$ x^2 - y^2 = xy + y $$
which isn't extraordinarily helpful.  If it had been x^2 + y^2, that would have been very helpful.