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
\(\displaystyle L(x,y)=\frac{1}{2}(\alpha(y^2)+x^2)+ \lambda[(x)+\theta (y)-\theta(z)-w]\)
\(\displaystyle \frac{\partial(L)}{\partial(x)}=x+\lambda=0 \)
\(\displaystyle \frac{\partial(L)}{\partial(x)}=\alpha(y)+(\theta)(\lambda)=0 \)
\(\displaystyle \frac{\partial(L)}{\partial(\lambda)}=(x)+\theta (y)-\theta(z)-w=0 \)
\(\displaystyle \lambda=-x \)
\(\displaystyle \alpha(y)=-(\theta)(\lambda)\)
\(\displaystyle \alpha(y)=(\theta)(x)\)
\(\displaystyle (y)=\frac{\theta}{\alpha}(x)\)
\(\displaystyle (x)+\theta (y)=\theta(z)+w\)
\(\displaystyle (x)+\theta (\frac{\theta}{\alpha}(x))=\theta(z)+w\)
>>>
\(\displaystyle (x)=\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]}\)
\(\displaystyle (y)=\frac{\theta}{\alpha}(\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]})\)
***
then I have to sub it in back into the original equation
\(\displaystyle f(x,y)=\frac{1}{2}(\alpha(y^2)+x^2)\)
\(\displaystyle f(x,y)=\frac{1}{2}(\alpha(\frac{\theta^2}{\alpha^2}\)\(\displaystyle (\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]})^2)+(\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]})^2)\)
\(\displaystyle f(x,y)=\frac{1}{2}[((\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]})^2)((\frac{\theta}{\alpha})^2+1)]\)
but how can I simply this further, have I made any mistakes?
Any help will be greatly appreciated!
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
\(\displaystyle L(x,y)=\frac{1}{2}(\alpha(y^2)+x^2)+ \lambda[(x)+\theta (y)-\theta(z)-w]\)
\(\displaystyle \frac{\partial(L)}{\partial(x)}=x+\lambda=0 \)
\(\displaystyle \frac{\partial(L)}{\partial(x)}=\alpha(y)+(\theta)(\lambda)=0 \)
\(\displaystyle \frac{\partial(L)}{\partial(\lambda)}=(x)+\theta (y)-\theta(z)-w=0 \)
\(\displaystyle \lambda=-x \)
\(\displaystyle \alpha(y)=-(\theta)(\lambda)\)
\(\displaystyle \alpha(y)=(\theta)(x)\)
\(\displaystyle (y)=\frac{\theta}{\alpha}(x)\)
\(\displaystyle (x)+\theta (y)=\theta(z)+w\)
\(\displaystyle (x)+\theta (\frac{\theta}{\alpha}(x))=\theta(z)+w\)
>>>
\(\displaystyle (x)=\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]}\)
\(\displaystyle (y)=\frac{\theta}{\alpha}(\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]})\)
***
then I have to sub it in back into the original equation
\(\displaystyle f(x,y)=\frac{1}{2}(\alpha(y^2)+x^2)\)
\(\displaystyle f(x,y)=\frac{1}{2}(\alpha(\frac{\theta^2}{\alpha^2}\)\(\displaystyle (\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]})^2)+(\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]})^2)\)
\(\displaystyle f(x,y)=\frac{1}{2}[((\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]})^2)((\frac{\theta}{\alpha})^2+1)]\)
but how can I simply this further, have I made any mistakes?
Any help will be greatly appreciated!
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