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Find lagrangian (please check work)

  1. Apr 17, 2014 #1

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    1. The problem statement, all variables and given/known data

    F(x, y) = 96xy - 4x
    subject to constraint of 11 = x + y

    Form the lagrangian.

    2. Relevant equations

    F(x, y) = 96xy - 4x
    subject to constraint of 11 = x + y

    3. The attempt at a solution

    My only question is solving 11 = x + y

    My book says the answer is:
    L = 96xy - 4x + λ(11 - x - y)

    But I got:
    L = 96xy - 4x + λ(x + y - 11)

    Just to confirm, both are correct because it just depends how you solve the constraint, no?
     
  2. jcsd
  3. Apr 17, 2014 #2

    Ray Vickson

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    It makes no difference: one λ will just have the opposite sign of the other.

    However, if does matter when you are doing post-optimality analysis. For example, you can use the value of λ to find the approximate change in the optimal value of F when the constraint changes to x+y = 11.1, for example. In that case you need to understand exactly which form of Lagrangian to use, or at least, how to apply either +λ or -λ in the analysis. It also matters whether you are maximizing or minimizing, and in your post you did not say which you were doing.
     
  4. Apr 17, 2014 #3

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    The goal was to find an optimal value subject to the constraint... Does it matter then, and how do you know which one to pick?

    Also, both partial derivatives would be correct, right?
     
  5. Apr 17, 2014 #4

    Ray Vickson

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    What is optimal? Maximum? Minimum?

    I don't know what you mean when you ask if both partials are correct; you did not give formulas for the partials, so I have no way to know if they are correct or not.
     
    Last edited: Apr 17, 2014
  6. Apr 17, 2014 #5

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    Sorry.

    I mean if you were NOT asked to find a maximum or minimum, and were ONLY asked to find the lagrangian and the three partial derivatives - would these two answers be correct regardless of how you solved the constraint function?
     
  7. Apr 17, 2014 #6

    Ray Vickson

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    No: you need to solve the equations correctly! I think maybe you meant to say "regardless of which form of Lagrangian you use". Then the answer would be yes, if you make no errors during solving.

    However: you don't really need to ask; you can just go ahead and do it both ways to see what you get. In fact, that would be faster than submitting a question and waiting for an answer!
     
    Last edited: Apr 17, 2014
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