1. 3.1: Increasing and Decreasing Functions

2. Definition
A function f is increasing on an interval if for any 2 numbers x1 and x2 in the interval x1<x2 implies f(x1) < f(x2)
A function f is decreasing on an interval if for any 2 numbers x1 and x2 in the interval x1<x2 implies f(x1) > f(x2)

3. Look at y = x2. What do you notice about the slopes?

4. Increasing/Decreasing Test
If f’(x) > 0 for all x in the interval (a, b), then f is increasing on the interval (a, b).
If f’(x) < 0 for all x in the interval (a, b), then f is decreasing on the interval (a, b).
If f’(x) = 0 for all x in the interval (a, b), then f is constant on the interval (a, b).

5. Use the I/D Test for y = x2.
What is the derivative? Where is the derivative positive? Where is the derivative negative?

6. Checkpoint 2 p. 185

7. Critical Numbers
If f is defined at c, then c is a critical number of f if f’(c) = 0 or f’(c) is undefined.

8. To Apply the I/D Test Find f’(x) Locate critical numbers
Set up a number line, test x-values in each interval

9. Example
Find the intervals on which f(x) =x3 – 12x is increasing and decreasing.

10. Example
Find the intervals on which is increasing and decreasing.

11. Now you try:
Determine the intervals on which the following functions are increasing/decreasing.

12. Last Example
Checkpoint 6 p. 190