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Lagrange Multipliers(just need confirmation)

  1. Oct 27, 2011 #1
    1. The problem statement, all variables and given/known data
    Use Lagrange multipliers to find the max and min values of the function subject to the given constraints:

    f(x1,x2,...,xn) = x1 + x2 + ... + xn
    constraint: (x1)^2 + (x2)^2 + ... (xn)^2 = 1


    3. The attempt at a solution

    fo x1 to xn values, x must equal 1/sqrt(n) in order to equal 1. [ g(x)=k --> the constraint ]

    so (1/sqrt(n))^2 + (1/sqrt(n))^2 + (1/sqrt(n))^2 + ETC = 1

    right? so how else could i answer that? how would a give a value for the min/max?
     
  2. jcsd
  3. Oct 27, 2011 #2

    ehild

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    The sum of xi2 has to be equal to 1. So you have to solutions xi=1/√n or xi=-1/√n.

    ehild
     
  4. Oct 27, 2011 #3
    but what about the max/min .... or did i technically answer that?
     
  5. Oct 27, 2011 #4

    ehild

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    What is the value of the function f if all xi-s are 1/√n ? and when all xi=-1/√n ?

    ehild
     
  6. Oct 27, 2011 #5
    always 1 ... so there is no max or min ? its just flat?
     
  7. Oct 27, 2011 #6

    ehild

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    f = ∑ xi. Why is it 1???????

    ehild
     
  8. Oct 27, 2011 #7
    ...i dont know ....
     
  9. Oct 27, 2011 #8

    ehild

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    Say n=2. f(x1,x2)=x1+x2. The constraint is x12+x22=1.
    You found that the extrema are when x1=x2=1/√2 or x1=x2=-1/√2. Substitute back to f:

    f=x1+x2=1/√2+1/√2=2/√2 or

    f=x1+x2=-1/√2+(-1/√2)=-2/√2

    Are the values of f equal to 1?

    ehild
     
  10. Oct 27, 2011 #9

    Ray Vickson

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    Is the function x1 + x2 + ...+ xn constant for all (x1, x2, ...,xn) on the sphere? Look at the cases of n = 2 and n = 3.

    RGV
     
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