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Lagrange multiplier problem

  1. Jul 28, 2011 #1
    1. Assume we have function V(x,y,z) = 2x2y2z = 8xyz and we wish to maximise this function subject to the constraint x^2+Y^2+z^2=9. Find the value of V at which the max occurs



    2. Function: V(x,y,z) = 2x2y2z = 8xyz
    Constraint: x^2+Y^2+z^2=9




    3. So far I have gone
    Φ= 8xyz + λ(x^2+y^2+z^2 - 9)

    ∂Φ/∂x = 8yz + 2λx = 0 equation 1
    ∂Φ/∂y = 8xz + 2λy = 0 equation 2
    ∂Φ/∂z = 8xy + 2λz = 0 equation 3
    x^2 + y^2 + z^2 = 9

    so then I have multiplied equation 1 by x, 2 by y, and 3 by z

    so im left with
    8xyz + 2λx^2 = 0
    8xyz + 2λy^2 = 0
    8xyz + 2λz^2 = 0

    add these together and
    24xyz +2λ(x^2 + y^2 + z^2)
    I know that x^2 + y^2 + z^2 =9 and v=8xyz so
    3V=-18λ
    V=-6λ

    Now I have no idea how to go about the next stage, I am struggling to rearrange the equations to get an answer.
     
  2. jcsd
  3. Jul 28, 2011 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    Your equations 8xyz + 2*lambda*x^2 = 0, etc., imply that x^2, y^2 and z^2 are equal, so it is easy to get them. Of course, x, y and z are thus determined only up to a +- sign, so you still need to think a bit about which choices make sense.

    RGV
     
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