In this section, we use the derivative to determine intervals on which a given function is increasing or decreasing. We will also determine the local extremes of the function.

Increasing and Decreasing Functions

     

The derivative is used to determine the intervals where a function is either increasing or decreasing. The following theorem is a direct consequence of the cornerstone, Mean Value Theorem (section 3.5).

     

(problem 1a) Determine whether the function is increasing or decreasing on .
increasing decreasing
(problem 1b) Determine whether the function is increasing or decreasing on .
increasing decreasing
(problem 2a) Determine whether the function is increasing or decreasing on .
increasing decreasing
(problem 2b) Determine whether the function is increasing or decreasing on .
increasing decreasing
(problem 3a) The derivative of is .
Is increasing or decreasing on the interval ?
increasing decreasing
(problem 3b) The derivative of is .
Is increasing or decreasing on the interval ?
increasing decreasing
(problem 4a) The derivative of is .
Is increasing or decreasing on the interval ?
increasing decreasing
(problem 4b) The derivative of is .
Is increasing or decreasing on the interval ?
for all values of
increasing decreasing
(problem 5a) The derivative of is .
Is increasing or decreasing on the interval ?
increasing decreasing
(problem 5b) The derivative of is .
Is increasing or decreasing on the interval ?
for all values of
increasing decreasing
(problem 6) The derivative of is .
Is increasing or decreasing on the interval ?
increasing decreasing
(problem 7) Determine intervals on which is increasing.
Select all that apply:
(problem 8) Determine intervals on which is increasing.
Select all that apply:

The work that was done in the previous example can actually give us slightly more information about . We can determine the local extremes of .

Critical numbers can help us find the location and the nature of local extremes and the next theorem tells us how.

(problem 9a) Find the local extremes of the function
Find the critical numbers of
Find intervals where is positive/negative
Use the First Derivative Test to determine the local extremes.

If there are none, type “none”.
has a local maximum at
has a local minimum at

(problem 9b) Find the local extremes of the function
Find the critical numbers of
Find intervals where is positive/negative
Use the First Derivative Test to determine the local extremes.

If there are none, type “none”.
has a local maximum at
has a local minimum at

(problem 9c) Find the local extremes of the function
Find the critical numbers of
Find intervals where is positive/negative
Use the First Derivative Test to determine the local extremes.

If there are none, type “none”.
has a local maximum at
has a local minimum at

(problem 9d) Find the local extremes of the function
Find the critical numbers of
Find intervals where is increasing/decreasing
Use the First Derivative Test to determine the local extremes.

If there are none, type “none”.
If there is more than one local extreme, list them in ascending order.
has a local maximum at
has a local minimum at
and at

(problem 9e) Find the local extremes of the function
Find the critical numbers of
Find intervals where is positive/negative
Use the First Derivative Test to determine the local extremes.

If there are none, type “none”.
has a local maximum at
has a local minimum at

(problem 10a) Find the local extremes of the function
Find the critical numbers of
Find intervals where is positive/negative
Use the First Derivative Test to determine the local extremes.

If there are none, type “none”.
has a local maximum at
has a local minimum at

(problem 10b) Find the local extremes of the function
Find the critical numbers of
Find intervals where is positive/negative
Use the First Derivative Test to determine the local extremes.

If there are none, type “none”.
has a local maximum at
has a local minimum at

(problem 10c) Find the local extremes of the function
Find the critical numbers of
Find intervals where is positive/negative
Use the First Derivative Test to determine the local extremes.

If there are none, type “none”.
has a local maximum at
has a local minimum at

(problem 11) Use the graph of shown below to determine where is increasing/decreasing and find the local extremes.

is increasing on the interval(s):

and

The local extremes of are:

local min at local min at and local max at local max at and local min at

Here are some detailed, lecture style videos on inc/dec functions:
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