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 each year. How should the items be priced to maximize profit?

Seems like they want to solve this using Lagrange multipliers (is it even possible?)

$P(x,y)=500(y-x)(x-40)+(45000+500(x-2y))(y-60)$

What could be the constraint?