Chapter 2.3, Problem 73E

Profit-loss analysis. Use the revenue cost and cost function from problem $71$ :

$\begin{array}{l}R\left(x\right)=x\left(75-3x\right)\text{ Revenue function}\\ C\left(x\right)=125+16xCost\text{function}\end{array}$

where $x$ is in millions of chips, and $R\left(x\right)$ and $C\left(x\right)$ are in millions of dollars. Both functions have domain $1\le x\le 20$.

(A) Form a profit function $P$, and graph $R,C,\text{ and }P$ in the same rectangular coordinate system.

(B) Discuss the relationship between the intersection points of the graphs of $R$ and $C$ and the intercepts of $P$.

(C) Find the $x$ intercepts of $P$ and the break-even points to the nearest thousand chips.

(D) Find the value of $x$ (to the nearest thousand chips) that produces the maximum profit. Find the maximum profit (to the nearest thousand dollars), and compare with problem $69\text{B}$.