# Maxima and Minima of two-variable functions word problem

1. Feb 13, 2013

### carleon

1. Problem:
A manufacturer makes two models of an item, standard and deluxe. It costs $40 to manufacture the standard model, and$60 for the deluxe. A market research firm estimates that if the standard model is priced at x dollars, and the deluxe at y dollars, then the manufacturer will sell 500(y-x) of the standard items and 45000 + 500(x-2y) of the deluxe items each year. How should the items be priced to maximize profit?

2. Relevant equations: none

3. The attempt at a solution:
I have f1x = -500, f1y = 500, and f2x = 500 and f2y = -1000
But none of these equations have critical points, so I know I'm supposed to check the boundaries next, but I don't know how to find them, or the absolute max. Also, never done this with two separate equations before, and I think that just means that I should do them seperately and use the points they have in common, but I'm not sure.
Thanks for the help!

2. Feb 13, 2013

### vela

Staff Emeritus
You're looking at the wrong functions. You want to maximize the profit p(x,y), so you're looking for the critical point of p(x,y). Start by writing down an expression for p(x,y).

3. Feb 14, 2013

### carleon

Thanks for responding! So I wrote an equation, which simplifies to P(x,y) = 45000-40x-560y, but I still have the same problems, where, fx = -40 and fy = -560. So how do I find the critical points?

4. Feb 14, 2013

### Ray Vickson

How did you get your P(x,y)---show the steps! This is important, because your P(x,y) is seriously wrong, and unless you show how you got it nobody can give you any helpful hints.