1. Increasing and Decreasing Functions
The First Derivative Test

2. Increasing and Decreasing Functions
Definitions of Increasing and Decreasing Functions

3. Test for Increasing and Decreasing Functions
Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b).
1. If f’(x) > 0 (positive) for all x in (a, b), then f is increasing on [a, b].
2. If f’(x) < 0 (negative) for all x in (a, b), then f is decreasing on [a, b].
3. If f’(x) = 0 for all x in (a, b), then f is constant on [a, b]. (It is a critical number.)

4. Determining intervals on which f is increasing or decreasing
Example:
Find the open intervals on which
is increasing or decreasing.

5. Guidelines for finding intervals
Let f be continuous on the interval (a, b). To find the open intervals on which f is increasing or decreasing, use the following steps:
(1) Locate the critical number of f in (a, b), and use these numbers to determine test intervals
(2) Determine the sign of f’(x) at one test value in each of the intervals
(3) If f’(x) is positive, the function is increasing in that interval. If f’(x) is negative, the function is decreasing in that interval.

6. Increasing Function

7. Decreasing Function

8. Constant Function

9. Strictly Monotonic
A function is strictly monotonic on an interval if it is either increasing on the entire interval or decreasing on the interval.

10. Not Strictly Monotonic

11. The First Derivative Test
Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows.
(1) If f’(x) changes from negative to positive at c, then f(c) is a relative minimum of f.
(2) If f’(x) changes from positive to negative at c, then f(c) is a relative maximum of f.
(3) If f’(x) does not change signs at c, then f(c) is neither a relative maximum nor a minimum.

12. First Derivative Test
On-line Video Help
Examples
More Examples

13. First Derivative Test Applications
Getting at the Concept & # 57 p. 182
The profit P (in dollars) made by a fast-food restaurant selling x hamburgers is
Find the open intervals on which P is increasing or decreasing

14. First Derivative Test Applications
On-line Applications