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

  1. #1
    New Member
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    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. #2
    Elite Member
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    Jun 2007
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    17,481
    Quote Originally Posted by Lichi View Post
    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!!
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  3. #3
    Elite Member
    Join Date
    Apr 2005
    Location
    USA
    Posts
    9,276
    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.
    "Unique Answers Don't Care How You Find Them." - Many may have said it, but I hear it most from me.

  4. #4
    New Member
    Join Date
    Dec 2017
    Posts
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    Quote Originally Posted by tkhunny View Post
    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.

    Thank you both for your answers.

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