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LaGrange Multipliers

  1. Apr 16, 2009 #1
    1. The problem statement, all variables and given/known data

    Consider the problem of finding the points on the surface [tex]xy+yz+zx=3[/tex] that are closest to the origin.

    1) Use the identity [tex](x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)[/tex] to prove that [tex]x+y+z[/tex] is not equal to 0 for any point on the given surface.

    2) Use the method of Lagrange multipliers to find a system of four equations in [tex]x,y,z[/tex] and [tex]\lambda[/tex] whose solutions will give the closest points.

    3) Find the points on [tex]xy+yz+zx=3[/tex] that are closest to the origin.

    2. Relevant equations
    3. The attempt at a solution

    I'm clueless on what to do for the 1st part (although I imagine it's actually something simple), but I think I have the second part down. Problem is, I think I probably need to use the 1st part for the 3rd somehow.

    For the second part, I found the system of four equations to be:

    [tex]2x=\lambda(y+z)[/tex]
    [tex]2y=\lambda(x+z)[/tex]
    [tex]2z=\lambda(x+y)[/tex]
    [tex]xy+yz+zx=3[/tex]
     
  2. jcsd
  3. Apr 16, 2009 #2
    For part 1: HINT - Use the fact that for any point on the surface xy + yz + xz = 3

    For part 3: HINT - Add the left sides of the first three equations and their right sides to make a new equation (then use part 1)
     
  4. Apr 16, 2009 #3

    Hurkyl

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    You can use the first part, even if you don't know why the first part is true. :tongue:

    By the way, part 3 is just a system of equations, solve it the way you normally would. The first part doesn't really affect that -- all ithe first part does is let you make a simplification along the way.
     
  5. Apr 16, 2009 #4
    I feel so stupid, I'm failing at solving this simple system, yet I am pretty sure I know what I'm going to end up with. I'm sure it will be something like [tex] x+y=-z[/tex] that I end up with, because than that would say [tex]x+y+z=0[/tex] which isn't true, which proves that [tex]x=y[/tex].

    EDIT: Nevermind, it's solved. I don't know why I was blanking on solving the system.
     
    Last edited: Apr 17, 2009
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