Chapter 7.3, Problem 53E

To determine

To prove: If the company’s profit is maximized when $x=a$, $y=b$, then $\frac{\partial C}{\partial x}(a,b)=p_1$ and $\frac{\partial C}{\partial y}(a,b)=p_2$, where $\$p_1$ and $\$p_2$ are the price per unit of the products $\text{I}$ and $\text{II}$ respectively and $C(x,y)$ be the cost of producing $x$ units of product $\text{I}$ and $y$ units of product $\text{II}$.