Chapter 2.7, Problem 3MP

Matched Problem 3 Refer to the revenue and profit functions in Example 3.

1. (A)   Find R′(3,000) and R′(6,000). Interpret the results.
2. (B)   Find P′(2,000) and P′(7,000). Interpret the results.

EXAMPLE 3 Production Strategy A company’s market research department recommends the manufacture and marketing of a new headphone. After suitable test marketing, the research department presents the following price–demand equation:

$x=10,000-1,000px\text{is}\text{demand}\text{at}\text{price}p.$    (1)

In the price–demand equation (1), the demand x is given as a function of price p. By solving (1) for p (add 1,000p to both sides of the equation, subtract x from both sides, and divide both sides by 1,000), we obtain equation (2), in which the price p is given as a function of demand x:

$p=10-0.001x$    (2)

where x is the number of headphones that retailers are likely to buy at $p per set.

The financial department provides the cost function

$C(x)=7,000+2x$    (3)

where $7,000 is the estimate of fixed costs (tooling and overhead) and $2 is the estimate of variable costs per headphone (materials, labor, marketing, transportation, storage, etc.).