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Max min, Lagrange's multiplier question

  1. Dec 23, 2006 #1
    1. The problem statement, all variables and given/known data
    [​IMG]


    2. Relevant equations
    possibly Lagrange's multiplier..


    3. The attempt at a solution
    treating S = f(x1, x2, ... , x2006) = x1 * 1^1/3 + x2 * 2^1/3 + ... + x2006 * 2006^1/3

    and constrain G(x1, x2 ... x2006) = x1 ^ 3/2 + x2 ^ 3/2 + ... + x2006 ^ 3/2 - (2^1/2 / (2006^1/2 * 2007 ^ 1/2)) = 0

    then solve G(x...) = 0
    gradient(f) = lambda * gradient(G) , which isn't all that clear what to do next.. any hints ?
     
  2. jcsd
  3. Dec 24, 2006 #2

    HallsofIvy

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    It isn't clear what to do next? How about completing that equation:
    [itex]\nabla S= \lambda \cdot \nabla G[/itex]? (S, not f)
    What is [itex]\nabla S[/itex]? What is [itex]\nabla G[/itex]? That should give you 2007 linear equations for x1, x2, . . . , x2006 and [itex]\lambda[/itex]. Fortunately they are almost all separated and solving just a few should give you the general formula.
     
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