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)-2λ(x-1/2)
0 = Ly = 2(y-1)-2λ(y-1/2)
0 = Lλ = -[(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"?
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)-2λ(x-1/2)
0 = Ly = 2(y-1)-2λ(y-1/2)
0 = Lλ = -[(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"?
