The document discusses how to determine if a function is increasing or decreasing on an interval using the derivative. It states that if the derivative is positive on an interval, the function is increasing on that interval, and if the derivative is negative, the function is decreasing. It provides steps to determine where a function is increasing or decreasing: 1) take the derivative, 2) find critical points where the derivative is 0 or undefined, 3) plot critical points to get intervals, 4) check sign of derivative in intervals. An example problem demonstrates finding the intervals where a cubic function is increasing or decreasing.

1. Section 5.1
Increasing and Decreasing Functions

2. Objectives
Upon completion of this lesson, you should be
able to:
• find where functions are increasing or
decreasing

3. Increasing/Decreasing
Functions
So far, we have only been able to determine if a
function is increasing or decreasing by plotting
points to graph the function.
Now that we know how to find the derivative of a
function, we will learn how the derivative can be
used to determine the intervals where a function is
increasing or decreasing.

4. Increasing/Decreasing
Functions
Remember, the derivative of a function represents
the slope of the tangent line at a particular point on
the graph.
So, if the derivative is positive on an open interval
(a, b), then the slope of the tangent line is positive,
which means the function is increasing on the
interval (a, b).
So, if the derivative is negative on an open interval
(a, b), then the slope of the tangent line is negative,
which means the function is decreasing on the
interval (a, b).

5. Increasing/Decreasing
Functions
A function f is increasing on (a, b) if f (x1) < f (x2)
whenever x1 < x2.
A function f is decreasing on (a, b) if f (x1) > f (x2)
whenever x1 < x2.
Increasing Decreasing Increasing

6. Increasing/Decreasing/Constant
Functions
If f ′( x ) > 0 for each value of x in an interval ( a, b ) ,
then f is increasing on ( a, b ).
If f ′( x ) < 0 for each value of x in an interval ( a, b ) ,
then f is decreasing on ( a, b ).
If f ′( x ) = 0 for each value of x in an interval ( a, b ) ,
then f is constant on ( a, b ).

7. Example
In the given graph of the function f(x), determine the
interval(s) where the function is increasing,
decreasing, or constant.

8. Example
Solution:
Looking at the graph from left to right, we would
have the following three intervals.
The function is decreasing on the interval (-4, -2)
The function is increasing on the interval (-2, 0)
The function is decreasing on the interval (0, 2)

9. Critical Numbers
In order to find the intervals where a function is
increasing, decreasing, or constant without first
graph the function, we must find what are called
critical numbers.
The critical numbers are those contained in the
domain of f(x) and which make the first derivative
equal to zero or undefined.

10. Critical Points of f
A critical point of a function f is a point in the
domain of f where
f ′( x) = 0 or f ′( x) does not exist.
(horizontal tangent lines, vertical tangent lines
and sharp corners)

11. Increasing/Decreasing
Functions
Steps in determining where a function is increasing or
decreasing:
1. Find the derivative of the given function.
2. Locate any critical numbers by seeing where the derivative is
either zero or undefined.
3. Plot the critical numbers on a number line to determine the
open intervals.
4. Select a test point in each interval and evaluate the derivative
at this point.
5. Use the sign of the derivative in each interval to determine
whether it is increasing or decreasing.

12. Example
Determine the intervals where f ( x) = x 3 − 6 x 2 + 1
is increasing and where it is decreasing.
f ′( x) = 3x 2 − 12 x
3x 2 − 12 x = 0
3 x ( x − 4) = 0
3x = 0 or x − 4 = 0
x = 0, 4
+ - +
0 4
f is decreasing
f is increasing
on ( 0, 4 )
on ( −∞, 0 ) ∪ ( 4, ∞ )