About increasing and decreasing functions and their critical numbers.

1. Suppose that a frog is about to
jump 3 steps up in the stairs a day
and the following day the frog will
jump 2 steps down the stairs. Now,
how many days will the frog reach
the top of the stair if it is 6 steps up
to reach the top?

2. -1-2-3-4-5-6 1 2 3 4 5 6

4. Increasing and Decreasing Functions
Definition:
If the graph of a function rises from left to
right over an interval I, it is said to be increasing
over an interval I. If the graph drops from left to
right over an interval I, it is said to be decreasing
over I.

5. Mathematically,
A function is increasing over I, if for every
𝑥1and 𝑥2 in I,
𝑥1 < 𝑥2, f(𝑥1) < f(𝑥2)
A function is decreasing on an interval I, if
for every 𝑥1 and 𝑥2 in I,
𝑥1 < 𝑥2, f(𝑥1) > f(𝑥2)

6. Increasing Functions

7. Decreasing Functions

8. Derivatives can be used to determine
whether a function is increasing or
decreasing:
 if 𝑓′
𝑥 > 0, for all 𝑥 in an interval I, then f is
increasing on I.
 if 𝑓′
𝑥 < 0, for all 𝑥 in an interval I, then f is
decreasing on I.
 if 𝑓′
𝑥 = 0, for all 𝑥 in an interval I, then f is
constant on I.

9. Checkout this graph.
You've got an ant climbing on the graph.
Not just any ant...

10. He's kind of a
pathetic math
superhero.
Pierre the Mountain Climbing Ant!

11. For this, the rule is that Pierre only
crawls from left to right…..

12. If Pierre is climbing uphill, then the
graph is increasing:
So, our graph is
increasing on:

13. If Pierre is going downhill, then the
graph is decreasing:
So, our graph is decreasing on:

14. When Pierre is standing right ON x = -3 ...
He's not going uphill
or downhill. He's just
standing there!

15. What if Pierre is walking on ?
That line is horizontal (slope of 0).
He's not going uphill or downhill,
so the graph is not increasing or decreasing there.

16. Critical Numbers
Definition:
The critical numbers for a function f are
those numbers c in the domain of f for which
𝑓′
𝑐 = 0 or 𝑓′
𝑐 does not exist.
A critical point is a point whose 𝑥 -
coordinate is the critical number c, and whose
𝑦-coordinate is 𝑓(𝑐).

18. APPLYING THE TEST
Locate the critical numbers for f on a number
line, as well as any points where f is undefined.
These points determine several open intervals.
Choose a value of 𝑥 in each of the intervals
determined in Step 1. Use these values to decide
whether 𝑓′(𝑥) > 0 or 𝑓′(𝑥) < 0 in that interval.
Use the test above to decide whether f is
increasing or decreasing on the interval.

19. Example 1:
Given 𝑓 𝑥 = 𝑥3
+ 6𝑥2
+ 9𝑥
a) Find the open interval where the given function
is increasing or decreasing.
b) Locate all points where the tangent line is
horizontal.
c) Graph the function.

20. Seatwork 
Given 𝑓 𝑥 = 2𝑥3
− 15𝑥2
+ 36𝑥 − 24
a) Find the open intervals where the given
function is increasing or decreasing.
b) Locate all points where the tangent line is
horizontal.
c) Graph the function.

21. Assignment 
DIRECTION:
Answer 5.1 Exercises on your book (Calculus
with Applications, 9th Edition) nos. 18, 19, 20 and
21 page 314.