# Langrange Multiplier Method

• Oct 19th 2010, 11:33 AM
Brokensound723
Langrange Multiplier Method
Need help using the Lagrange Multiplier Method on this problem. I could easily optimize this with direct substitution but I need to know how to solve it using this method.

$\displaystyle z=f(x,y)=3(x-5)^2+4y+1307$

The Function is constrained by $\displaystyle x+y=144$

so I have $\displaystyle L(x,y)=3(x-5)^2+4y+1307-lambda(x+y-144)$

Then $\displaystyle Lx(x,y)=6(x-5)-lambda=0$
$\displaystyle Ly(x,y)=4-lambda=0$
$\displaystyle g(x,y)=x+y-144=0$

• Oct 19th 2010, 12:49 PM
pickslides
Quote:

Originally Posted by Brokensound723

$\displaystyle z=f(x,y)=3(x-5)^2+4y+1307$

The Function is constrained by $\displaystyle x+y=144$

so I have $\displaystyle L(x,y)=3(x-5)^2+4y+1307-lambda(x+y-144)$

I get

$\displaystyle L(x,y,\lambda)=3(x-5)^2+4y+1307+\lambda (x+y-144)$

with

$\displaystyle L_x=6(x-5)+\lambda =0$

$\displaystyle L_y=4+\lambda =0$

$\displaystyle L_{\lambda}=x+y-144=0$

Now solve the system

$\displaystyle 6(x-5)+\lambda =0$ ...(1)

$\displaystyle 4+\lambda =0$ ...(2)

$\displaystyle x+y-144=0$ ...(3)