# Lagrangian function.

• Sep 27th 2009, 08:53 PM
jebckr
Lagrangian function.
f(x,y)=2x^2+y^2 subject to the constraint of x+y=1.

I understand how to set it up and how to find f with respect to x y and L.

fx=4x+L
fy=2y+L
fL=x+y-1

So, again...I understand the cal part of it, but fail to remember the basic allgebra part. Please solve for x, y, and L (ie the critical point)
• Sep 27th 2009, 09:06 PM
pickslides
set $\displaystyle f_x, f_y$ and $\displaystyle f_L = 0$ then solve the system.
• Sep 27th 2009, 09:14 PM
jebckr
I know to set them equal to 0, but I'm not exactly sure how to solve from there. This may actually be the wrong section since this is just algebra...
• Sep 27th 2009, 09:39 PM
pickslides
$\displaystyle f_x=4x+L=0$ ...(1)
$\displaystyle f_y=2y+L=0$ ...(2)
$\displaystyle f_L=x+y-1=0$ ...(3)

Using (1) and (2)

$\displaystyle 4x+L=2y+L$

Therefore

$\displaystyle 4x=2y$

$\displaystyle 2x=y$

Putting this into (3)

$\displaystyle x+2x-1=0$

$\displaystyle 3x-1=0$

$\displaystyle x=\frac{1}{3}$

Can you solve it from here?
• Sep 27th 2009, 09:40 PM
jebckr
Oh, I feel dumb now... Thank you!