1. Increasing/ Decreasing Functions
Section 4.2a

2. Definition: Increasing/Decreasing Functions
Let be a function defined on an interval I and
let and be any two points in I.
(a) f increases on I if
As x gets bigger, y gets bigger…
(b) f decreases on I if
As x gets bigger, y gets smaller…
A function that is always increasing or decreasing on
a particular interval is monotonic on that interval

3. The “Do Now” Writing: How can the derivative help in identifying
when a function is increasing or decreasing?
Let be continuous on and differentiable on
If at each point of , then
increases on
(b) If at each point of , then
decreases on

4. “Seeing” this new tool…
Consider the function
Next, the derivative:
First, look at the graph: y
on , so
the function is decreasing on
on , so
the function is increasing on x

5. Guided Practice Minimum of at
Use analytic methods to find (a) the local extrema, (b) the
intervals on which the function is increasing, and (c) the intervals
on which the function is decreasing.
The derivative:
There are no endpoints, and the only critical point occurs at:
Minimum of at

6. Guided Practice
Use analytic methods to find (a) the local extrema, (b) the
intervals on which the function is increasing, and (c) the intervals
on which the function is decreasing.
The derivative:
(b) Since on ,
is increasing on
(c) Since on ,
is decreasing on

7. Guided Practice
Use analytic methods to find (a) the local extrema, (b) the
intervals on which the function is increasing, and (c) the intervals
on which the function is decreasing.
The derivative:
Since the derivative is never zero and is undefined only
where is undefined, there are no critical points. Also, the
domain has no endpoints. Therefore has no local extrema.

8. Guided Practice
Use analytic methods to find (a) the local extrema, (b) the
intervals on which the function is increasing, and (c) the intervals
on which the function is decreasing.
The derivative:
(b) on
So, the function is increasing on
(c) on
So, the function is decreasing on

9. Guided Practice
Find (a) the local extrema, (b) the intervals on which the function
is increasing, and (c) the intervals on which the function is
decreasing.
First, graph
the function…
The derivative:
Critical Points:
x = –2, x = 0
The local extrema can occur at the critical points, but the
graph shows that no extrema occurs at x = 0. There is a local
(and absolute) minimum at

10. Guided Practice
Find (a) the local extrema, (b) the intervals on which the function
is increasing, and (c) the intervals on which the function is
decreasing.
(b) Since on the intervals: …
And since is continuous at x = 0…
is increasing on
(c) Since on the interval: …
is decreasing on

11. Guided Practice
Find (a) the local extrema, (b) the intervals on which the function
is increasing, and (c) the intervals on which the function is
decreasing.
First, graph
the function…
The derivative:

12. Guided Practice
Find (a) the local extrema, (b) the intervals on which the function
is increasing, and (c) the intervals on which the function is
decreasing.
Since the derivative is never zero and is undefined only where
the function is undefined, there are no critical points. Since there
are no critical points and the domain includes no endpoints, the
function has no local extrema.
(b) Since the derivative is never positive, the function is not
increasing on any interval.
(c) Since the derivative is whenever it is defined, the function is
decreasing on: