1. ## Lagrange Multipliers

A manufacturing company has three plants, I, II and III, which produce x, y, and z units, respectively, of a
certain product. The annual revenue from this production is given by
R(x, y, z) = 6xyz^2 − 400000(x + y + z)
If the company is to produce 1000 units anually, use Lagrange Multipliers to determine how it should allocate
production among the three plants so as to maximize its revenues.

2. We have $\displaystyle f(x,y,z)=6xyz^{2}-400,000(x+y+z)$ subject to the constraint g(x,y,z)=x+y+z-1000[/tex]

$\displaystyle {\nabla}f(x,y,z)={\lambda}{\nabla}g(x,y,z)$

$\displaystyle f_{x}=6yz^{2}-400,000={\lambda}$....[1]

$\displaystyle f_{y}=6xz^{2}-400,000={\lambda}$...[2]

$\displaystyle f_{z}=12xyz-400,000={\lambda}$...[3]

Equate [1] and [2]:

$\displaystyle 6yz^{2}=6xz^{2}; \;\ x=y$

Equate [1] and [3]:

$\displaystyle 6yz^{2}-400,000=12xyz-400,000$

$\displaystyle z=2x$

Sub into the constraint:

$\displaystyle x+x+2x=1000$

$\displaystyle 4x=1000$

$\displaystyle \boxed{x=250, \;\ y=250, \;\ z=500}$