A marketing survey indicates that t months after a new type of computerized air purifier is introduced to the market, sales will be generating profit at the rate of P'(t) thousand dollars per month, where

a. When is the rate of profitability positive and when is it negative? When is the rate increasing and when is it decreasing?

b. At what time t = t_m is monthly profit maximized? Find the net change in profit over the time period 0 ≤ t ≤ t_m.

c. It costs the manufacturer $100,000 to develop the purifier product, so P(0) = −100. Use this information along with integration to find P(t).

d. Sketch the graph of P(t) for t  ≥  0. A fad is a product that gains rapid success in the market, then just as quickly fades from popularity. Based on the graph P(t), would you call the purifiers a fad? Explain.

Transcribed Image Text:

P' (t) = 500[1.4 In(0.5t + 1)] t + 2