Extreme Values Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

Finding extreme values of functions and their locations (i.e., finding the largest or smallest values of some function f(x), and the value(s) of x where those extremes occur) is one of the classic uses of derivatives. Section 4.3 in our textbook discusses it, and this discussion is intended to help you make sense of that reading.

To start, look at the ideas behind the process. For example, suppose Geneseo Widget Works has some function that they believe predicts the number of widgets they’ll sell t days after the start of a new advertising campaign (t > 0). They want to know when they can expect the maximum number of sales. Although you don’t at this point know their exact function (they know it though), what process would you suggest Geneseo Widget Works follow in order to find the time at which sales will peak?