Maximizing Profit with two limited products

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?

• Hi and welcome to math.SE. Please use MathJax formatting to improve readability and increase your chances to get meaningful answers. – francescop21 Oct 15 '18 at 9:45

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.