Maxima and Minima of two-variable functions word problem

[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. Homework 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 separately and use the points they have in common, but I'm not sure.
Thanks for the help!

 

[vela]

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).

 

[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?

 

[Ray Vickson]

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?

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.