# Thread: lagrangian multiplier method help!

1. ## Lagrange multiplier method help!

Hi,

I am trying to work on this problem, I have already sett up the Lagrange equation, i need to solve for x and y

$L(x,y)=\frac{1}{2}(\alpha(y^2)+x^2)+ \lambda[(x)+\theta (y)-\theta(z)-w]$

$\frac{\partial(L)}{\partial(x)}=x+\lambda=0$
$\frac{\partial(L)}{\partial(x)}=\alpha(y)+(\theta) (\lambda)=0$
$\frac{\partial(L)}{\partial(\lambda)}=(x)+\theta (y)-\theta(z)-w=0$

$\lambda=-x$
$\alpha(y)=-(\theta)(\lambda)$
$\alpha(y)=(\theta)(x)$
$(y)=\frac{\theta}{\alpha}(x)$

$(x)+\theta (y)=\theta(z)+w$
$(x)+\theta (\frac{\theta}{\alpha}(x))=\theta(z)+w$
>>>
$(x)=\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]}$

$(y)=\frac{\theta}{\alpha}(\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]})$

***
then I have to sub it in back into the original equation

$f(x,y)=\frac{1}{2}(\alpha(y^2)+x^2)$
$f(x,y)=\frac{1}{2}(\alpha(\frac{\theta^2}{\alpha^2 }$$(\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]})^2)+(\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]})^2)$

$f(x,y)=\frac{1}{2}[((\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]})^2)((\frac{\theta}{\alpha})^2+1)]$

but how can I simply this further, have I made any mistakes?

Any help will be greatly appreciated!

2. ..

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