3.B. Increasing/Decreasing Intervals and Extrema & Critical Numbers

In calculus (and in life), much effort is devoted to determining the behavior of something. In calculus's case, the effort is put into determining the behavior of a function on an interval. Various questions about the behavior of functions on an interval include:
  • Does have a maximum value on said interval?
  • Does it have a minimum value?
  • Where is the function increasing?
  • Where is it decreasing?
Answers to these types of questions have tremendous implications to real world applications.

In this section, you will learn how derivatives can be used to relative extrema as either relative minima or relative maxima. First, it is important to define increasing and decreasing functions.

A function is increasing if, its graph moves up, and is decreasing if its graph moves down. A positive derivative implies that the function is increasing; a negative derivative implies that the function is decreasing; and a zero derivative on an entire interval implies that the function is constant on that interval.

Essential Questions
  • How is the first derivative related to relative extrema and intervals of increasing/decreasing?
  • How are critical numbers and extrema related?