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Homework Help: Max values for function of 3 variables with two constraints

  1. Mar 31, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the maximum values of f(x,y,z)=xy+xz+yz-4xyz subject to the constraints x+y+z=1 and x,y,z>or equal 0.

    2. Relevant equations

    3. The attempt at a solution
    see attachment
    I found fgrad and ggrad and set fgrad equal to lambda*ggrad and used x+y+z=1 for my system of equations. I am have difficulty solving for x,y,z,lambda. I've spent over half hour attempting to algebraically solve for the variables. Please direct as to how to solve this system of equations. I need help with this problem ASAP!! Thanks Dan

    Attached Files:

  2. jcsd
  3. Mar 31, 2009 #2
    Help Me It is very important I complete this problem. How do i solve the 4 equations?
  4. Mar 31, 2009 #3


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    Homework Helper

    hi dan, been a while since i've done these, but see if this helps, (and if you agree with it...)
    your equations
    [tex] y+z-4yz = \lambda [/tex]
    [tex] x+z-4xz = \lambda [/tex]
    [tex] y+x-4yx = \lambda [/tex]
    [tex] x + y + z = 1 [/tex]
    [tex] x, y, z > 0 [/tex]

    equating each one in terms of lamba and factoring gives, for the first one
    [tex] y+z-4yz = x+z -4xz [/tex]
    [tex] y(1-4z) = x(1-4z) [/tex]
    [tex] x(1-4y) = z(1-4y) [/tex]
    [tex] z(1-4x) = y(1-4x) [/tex]
    with the constraints
    [tex] x + y + z = 1 [/tex]
    [tex] x, y, z > 0 [/tex]
    hopefully thsi helps, though you should also check the boundary points too
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