Math 22 Increasing and Decreasing Functions

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Definitions of Increasing and Decreasing Functions

 A function  is increasing on an interval when, for any two numbers  and 
  in the interval,  implies 
 A function  is decreasing on an interval when, for any two numbers  and 
  in the interval,  implies 

Test for Increasing and Decreasing Functions

 Let  be differentiable on the interval .
 1. If  for all  in , then  is increasing on .
 2. If  for all  in , then  is decreasing on .
 3. If  for all  in , then  is constant on .

Critical Numbers and Their Use

 If  is defined at , then  is a critical number of  when  or when  is 

Exercises: Find critical numbers of


So, is critical number


In this case, we have critical number when is undefined, which is when . So critical number is

Increasing and Decreasing Test

 1. Find the derivative of .
 2. Locate the critical numbers of  and use these numbers to determine test intervals.
 That is, find all  for which  or  is undefined.
 3. Determine the sign of  at one test value in each of the intervals.
 4. Use the test for increasing and decreasing functions to decide 
 whether  is increasing or decreasing on each interval.

Exercises: Find the intervals of increasing and decreasing of


So, are critical numbers.
Hence, the test intervals are and
In each interval, choose a number and test on :
So, , so is increasing on
, so is decreasing on
, so is increasing on
Therefore, is Increasing:
Decreasing on

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