(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the extrema of the given function subject to the given constraint:

f(x,y)=x^{2}-2xy+2y^{2}, subject to x^{2}+y^{2}=1

2. Relevant equations

Lagrange Multipliers

3. The attempt at a solution

First, I defined the constraint to be g(x,y)=0, that is,

g(x,y)=x^{2}+y^{2}-1

I then set up the usual basic system of equations (I did not show my work for this part because I find it unnecessary):

x-y=[tex]\lambda[/tex]x

-x+2y=[tex]\lambda[/tex]y

x^{2}+y^{2}-1=0

(*Note: I reduced the first two equations by a factor of 2 for the sake of easier computing).

So now, I have three sets of equation and the MOST algebraic manipulation I could

do is the following:

y=x(1-[tex]\lambda[/tex])

x=y(2-[tex]\lambda[/tex])

x^2+y^2=1

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Another path I took was combining the first two equations, which got me:

[tex]\lambda[/tex](x+y)=y

Which got me no where.

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And now, I am stuck! Oh yes, I forgot to mention that I am not allowed to use polar

coordinates at all (since the constraint is a unit circle...) - I am only allowed to use Lagrange Multipliers. On another note, the answer is

very, very ugly (it is in decimals). Can anyone please show me how to figure this problem

out? Thanks!

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# Homework Help: Very Frustrating (or Easy) Lagrange Multipliers Problem

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