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Lagrange multipliers and partial derivatives

  1. Dec 13, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the point on 2x + 3y + z - 11 = 0 for which 4x^2 +y^2 +z^2 is a minimum

    2. Relevant equations



    3. The attempt at a solution

    Using lagrange multipliers I find:

    F = 4x^2 + y^2 + z^2 + l(2x + 3y + z)

    Finding the partial derivatives I get the three equations:

    df/dx = 8x + 2l
    df/dy= 2y + 3l
    df/dz= 2z + l

    This is where I am stuck, what are the next steps for solving the system of equations?
     
  2. jcsd
  3. Dec 13, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Well, first set them equal to 0! They are equivalent to [itex]x= -\lambda[/itex], [itex]2y/3-\lambda[/itex] and [itex]2z= -\lambda[/itex]. You can easily eliminate [itex]\lambda[/itex] by setting those equal to each other. And don't forget that you have 2x+ 3y+ z= 11 as a third equation.
     
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