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Lagrange Multiplier Question

  1. Feb 29, 2012 #1

    ElijahRockers

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

    f(x,y) = y2-x2, g(x,y) = x2/4 +y2=9

    2. Relevant equations

    [itex]\nabla f = \lambda \nabla g[/itex]

    [itex]-2x = \lambda \frac{x}{2}[/itex]
    [itex]2y = 2\lambda y[/itex]
    [itex]\frac{1}{4} x^2 + y^2 = 9[/itex]

    3. The attempt at a solution

    I arrived at the three equations above. So according to the first equation, lambda can equal -4. According to the second equation, it can equal 1. After this, I am algebraically lost. The x's and y's cancel themselves out from the first two equations. What does this mean?
     
  2. jcsd
  3. Feb 29, 2012 #2

    Dick

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    −2x=λx/2 means EITHER λ=(-4) OR x=0. You have to check both options.
     
  4. Feb 29, 2012 #3

    Ray Vickson

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    If x ± 0 then λ = -4, so in the second equation you must have y = 0.

    RGV
     
    Last edited: Feb 29, 2012
  5. Feb 29, 2012 #4

    Dick

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    What are you talking about?
     
  6. Feb 29, 2012 #5

    Ray Vickson

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    I had the typo λ = 4 instead of the correct λ = -4, but that still implies we need y = 0 to satisfy the second equation (which would be 2y = -8y).

    RGV
     
  7. Feb 29, 2012 #6

    ElijahRockers

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    Ok, well a lambda of -4 makes the other equation untrue. So lambda can not be -4 then, right?
     
  8. Mar 1, 2012 #7

    Dick

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    Yes, it can. lambda can be -4 if y is zero.
     
  9. Mar 1, 2012 #8

    ElijahRockers

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    Duh, ok, I think I got it. Thanks again.
     
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