0
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ Hi and welcome to math.SE. Please use MathJax formatting to improve readability and increase your chances to get meaningful answers. $\endgroup$ – francescop21 Oct 15 '18 at 9:45
0
$\begingroup$

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.

enter image description here

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.