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A bookstore can obtain a certain gift book from the publisher at a cost of $3 per book. The bookstore has been offering the book at a price of $15 per copy and, at this price, has been selling 200 copies a month. The bookstore is planning to lower its price to stimulate sales and estimates that for each $1 reduction in the price, 20 more books will be sold each month. At what price should the bookstore sell the book to generate the greatest possible profit?

Let x be the price of the book ( tex2html_wrap_inline25 ) and N be the number of books sold at that price. Then





To maximize P(x), find the derivative:


This always exists, and is zero when x=14. To see that this is where P(x) attains an absolute maximum, note that P''(x)=-40 is always negative on the domain, so P(x) is always concave down. Therefore the price of the book should be chosen to be $14.00.

Another way to do this problem would be to let x equal the amount of reduction in price. Then N=200+20x, C=3N=600+60x, tex2html_wrap_inline45 , and tex2html_wrap_inline47 . Maximizing P yields x=1, representing a $1 reduction in price from $15 to $14.

Carl Lee
Wed Nov 4 10:27:26 EST 1998