Increasing and Decreasing Intervals and Extrema

  • 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?