1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    lanedance

    User Avatar
    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]
    with
    [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]
    gives
    [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]
    and
    [tex] x, y, z > 0 [/tex]
    hopefully thsi helps, though you should also check the boundary points too
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook