Hi, I'm having trouble with the following question.(adsbygoogle = window.adsbygoogle || []).push({});

Q. Find the maximum and minimum of the function f(x,y) = x^2 + xy + y^2 on the circle x^2 + y^2 = 1.

I started off by writing:

Let g(x,y) = x^2 + y^2 then [tex]\nabla f = \lambda \nabla g,g\left( {x,y} \right) = 1[/tex]

[tex]

\Rightarrow 2x + y = 2\lambda x...(1)

[/tex]

[tex]

2y + x = 2\lambda y...(2)

[/tex]

[tex]

x^2 + y^2 = 2...(3)

[/tex]

I'm not sure how to solve this system of equations. I've got the impression that generally an explicit value for lamda is not needed. I'd normally start off by considering the possible cases.

Here, if x = 0 in (1) then from (2) I get y = 0 so that (x,y) = (0,0). However this contradicts (3) and it isn't on the circle so I ignore this point. But now I'm stuck. Which values of x or y should I try now? Any help would be good thanks.

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# Homework Help: Lagrange Multipliers

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