# Max values for function of 3 variables with two constraints

1. Mar 31, 2009

### jimbo71

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

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2. Mar 31, 2009

### jimbo71

Help Me It is very important I complete this problem. How do i solve the 4 equations?

3. Mar 31, 2009

### lanedance

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

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