# Thread: Finding x,y and λ values from partial derivatives

1. ## Finding x,y and λ values from partial derivatives

Hi. I've got a couple of problems which i can't continue solving because i get stuck and im not figuring a way to solve it.

Here's one of the problems (the other i might post it later):
I've got the following function: L(λ, x, y) = (x-1)^2+(y-1)^2-λ[(x-1/2)^2+(y-1/2)^2-1/2] and i've got to find the critical points for (x, y).

So i compute the derivative with respect to x, y and λ:

0 = Lx = 2(x-1)-2
λ(x-1/2)
0 = Ly = 2(y-1)-2λ(y-1/2)
0 = L
λ = -[(x-1/2)^2+(y-1/2)^2-1/2]

And it is at that point in which i dont know how to continue.

In the problem sheet/paper which i have somebody solved it and wrote:

From Lx and Ly
⇒ x-y - λ(x-y) = 0
Then he factorizes: (x-y)(1-
λ)=0
And comes to the conclusion that either x=y or
λ=1

The problem is that i don't figure out how he came to that.

Can someone help me find out why from Lx and Ly i can get to "
x-y - λ(x-y) = 0"?

2. Originally Posted by Lichi
Hi. I've got a couple of problems which i can't continue solving because i get stuck and im not figuring a way to solve it.

Here's one of the problems (the other i might post it later):
I've got the following function: L(λ, x, y) = (x-1)^2+(y-1)^2-λ[(x-1/2)^2+(y-1/2)^2-1/2] and i've got to find the critical points for (x, y).

So i compute the derivative with respect to x, y and λ:

0 = Lx = 2(x-1)-2
λ(x-1/2)
0 = Ly = 2(y-1)-2λ(y-1/2)
0 = L
λ = -[(x-1/2)^2+(y-1/2)^2-1/2]

And it is at that point in which i dont know how to continue.

In the problem sheet/paper which i have somebody solved it and wrote:

From Lx and Ly
⇒ x-y - λ(x-y) = 0
Then he factorizes: (x-y)(1-
λ)=0
And comes to the conclusion that either x=y or
λ=1

The problem is that i don't figure out how he came to that.

Can someone help me find out why from Lx and Ly i can get to "
x-y - λ(x-y) = 0"?

0 = Lx = 2(x-1)-2
λ(x-1/2) = 0 = Ly = 2(y-1)-2λ(y-1/2)

2(x-1)-2λ(x-1/2) = 2(y-1)-2λ(y-1/2)

Now use pencil and paper (and eraser too) and simplify it!!

3. You should try whatever methods are available to you to solve a system of equations.

In this case, your predecessor seem of have produced, via subtraction, Lx - Ly.

4. Originally Posted by tkhunny
You should try whatever methods are available to you to solve a system of equations.

In this case, your predecessor seem of have produced, via subtraction, Lx - Ly.
Ahh great.