Matched Problem 3 Refer to the revenue and profit functions in Example 3.
- (A) Find R′(3,000) and R′(6,000). Interpret the results.
- (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:
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:
where x is the number of headphones that retailers are likely to buy at $p per set.
The financial department provides the cost function
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.).
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