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Lagrange multiplier

  1. Nov 28, 2009 #1
    1. The problem statement, all variables and given/known data
    Use the Lagrange Multiplier method to find the maximum and minimum values of x2 − 2xy + 7y2 on the ellipse x2 + 4y2 = 1.

    2. Relevant equations
    Lagrange multiplier method

    3. The attempt at a solution
    L(x,y,z,λ) = x2 − 2xy + 7y2 - λ(x2 + 4y2 - 1)
    Find Lx, Ly, Lλ
    Then, solve for x and y?
     
    Last edited: Nov 28, 2009
  2. jcsd
  3. Nov 28, 2009 #2

    ideasrule

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    Where did you get the L(x,y,z,λ)? You don't need the z in there because we're dealing with functions of x and y, not functions of x, y, and z.

    Also, λ is a constant: you don't need to solve for Lλ. You don't need to solve for Lz either because there's no such variable as z. Just set Lx=0 and Ly=0 and solve. Don't forget to satisfy the initial constraint of x^2+4y^2=1!
     
  4. Nov 28, 2009 #3

    HallsofIvy

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    Just an added note: some people are taught to find [itex]L_\lambda[/itex]. Because if we are trying to extremize F(x) subject to the constraint G(x)= 0, we look at [itex]L= F(x)+ \lambda G(x)[/itex], [itex]L_{\lamba}= G(x)= 0[/itex] is just the constraint again.
     
  5. Nov 28, 2009 #4
    @ideasrule: Sorry, ignore the "z."
    @HallsofIvy: Thanks. And yes, that's how I learned it.

    Also, once I find the critical points, do I just plug them back into the original function I want to extremize to see if it's a max or min?
     
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