Section 5.1

Increasing and Decreasing Functions

Upon completion of this lesson, you should be able to:

• find where functions are increasing or decreasing

Objectives

Increasing/DecreasingFunctions

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.

Increasing/DecreasingFunctions

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).

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 IncreasingDecreasing

Increasing/DecreasingFunctions

Increasing/Decreasing/ConstantFunctions

.,on increasing is then

,, intervalan in of each valuefor 0 If

baf

baxxf

.,on decreasing is then

,, intervalan in of each valuefor 0 If

baf

baxxf

.,on constant is then

,, intervalan in of each valuefor 0 If

baf

baxxf

Example

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

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)

Example

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.

A critical point of a function f is a point in the domain of f where

exist.not does )(or 0)( xfxf (horizontal tangent lines, vertical tangent lines and sharp corners)

Critical Points of f

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.

Increasing/DecreasingFunctions

16)( 23 xxxf

xxxf 123)( 2 0123 2 xx

Determine the intervals where

0)4(3 xx

04or 03 xx4,0x

0 4

+ - +

f is increasing

on

is increasing and where it is decreasing.

,0 4,

f is decreasing

on 0,4

Example