Profit-loss analysis. Use the revenue and cost functions from Problem 67: where...

Profit-loss analysis. Use the revenue and cost functions from Problem 67:

where x is in millions of chips, and R(x) and C(x) are in millions of dollars. Both functions have domain 1 ≤ x ≤ 20.

(A) Form a profit function P, and graph R, C, 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 to the nearest thousand chips.

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

(E) Verify your conclusion in part (D) by finding 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 65B.

Reference: Problem 67

Break-even analysis. Use the revenue function from Problem 65 and the given cost function:

where x is in millions of chips, and R(x) and C(x) are in millions of dollars. Both functions have domain 1 ≤ x ≤ 20.

(A) Sketch a graph of both functions in the same rectangular coordinate system.

(B) Find the break-even points to the nearest thousand chips.

(C) For what values of x will a loss occur? A profit?

Reference: Problem 65

Revenue. The marketing research department for a company that manufactures and sells memory chips for microcomputers established the following price–demand and revenue functions:

where p(x) is the wholesale price in dollars at which x million chips can be sold, and R(x) is in millions of dollars. Both functions have domain 1 ≤ x ≤ 20.

(A) Sketch a graph of the revenue function in a rectangular coordinate system.

(B) Find the value of x that will produce the maximum revenue. What is the maximum revenue?

(C) What is the wholesale price per chip that produces the maximum revenue?