Chapter 2.3, Problem 74E

Profit-loss analysis. Use the revenue function from Problem $70$ and the given cost function:

$\begin{array}{l}R\left(x\right)=x\left(2,000-60x\right)\text{ Revenue}\text{function}\\ C\left(x\right)=4,000+500x\text{ Cost}\text{function}\end{array}$

where $x$ is thousands of computers, and $R\left(x\right)\text{and }C\left(x\right)$ are in thousands of dollars. Both functions have domain $1\le x\le 25$.

(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 $R\text{ and }C$ and the $x$ intercepts of $P$.

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

(D) Find the value of $x$ that produces the maximum profit. Find the maximum profit and compare with Problem $70\text{B}$.