Chapter 2.3, Problem 70E

Profit-loss analysis. Use the revenue and cost functions from Problem 66:

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

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 $R$ and $C$ and the $x$ intercepts of $P$.

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

(D) Refer to the graph drawn in part (A). Does the maximum profit appear to occur at the same value of $x$ as the maxi- mum revenue? Are the maximum profit and the maximum revenue equal? Explain.

(E) Verify your conclusion in part (D) by finding the value of $x$ that produces the maximum profit. Find the maxi- mum profit and compare with Problem 66B.