The Problem
A company uses two intermediate products to produce their end product. The profit when using x (in thousands) of product 1, and z (in hundreds) of product 2 is known to be
P = (x − 0.5)(z − 1)(11 − 2x − z) + 500
Both products are limited, we have x ≤ 6 and z ≤ 10.
I am trying to find out which combination of x and z will create the Maximum Profit.
Currently I have determined the first derivative as (z-1)(-4x-z+12)
I am unsure how to progress from here with two variables as well as the limit imposed on both variables. Does anyone know the procedure for this type of Profit Maximization Question?
Calling $f(x,z) = (x-0.5)(z-1)(11-2x-z)$ the relative minimum/maximum/saddle points obey the condition
$$ \frac{\partial f}{\partial x} = 4x (1-z)+z (13 -z)-12 = 0\\ \frac{\partial f}{\partial y} = x (x+z-6.5)-0.5 z+3 = 0 $$
Those solutions are shown in red, over the level contour map for $f(x,z)$. Now I leave to you the corresponding qualification as relative minimum/maximum/saddle points.