This isn't very complete but it does give the highlights of the section.

1. Section 2.9
What does f say about f ?
Math 1a
October 17, 2007
Announcements
Midterm I review session 10/21, 7:30pm in Hall D?

2. Outline
Increasing and Decreasing functions
Concavity and the second derivative

3. Definition
Let f be a function defined on and interval I . f is called
increasing if
f (x1 ) < f (x2 ) whenever x1 < x2
for all x1 and x2 in I .

4. Definition
Let f be a function defined on and interval I . f is called
increasing if
f (x1 ) < f (x2 ) whenever x1 < x2
for all x1 and x2 in I .
f is called decreasing if
f (x1 ) > f (x2 ) whenever x1 < x2
for all x1 and x2 in I .

5. Fact
If f is increasing on (a, b), then f (x) ≥ 0 for all x in (a, b)

6. Fact
If f is increasing on (a, b), then f (x) ≥ 0 for all x in (a, b)
If f is decreasing on (a, b), then f (x) ≤ 0 for all x in (a, b).

7. Fact
If f (x) > 0 for all x in (a, b), then f is increasing on (a, b).
If f (x) < 0 for all x in (a, b), then f is decreasing on (a, b).

8. Outline
Increasing and Decreasing functions
Concavity and the second derivative

9. Definition
A function is called concave up on an interval if f is
increasing on that interval.

10. Definition
A function is called concave up on an interval if f is
increasing on that interval.
A function is called concave down on an interval if f is
decreasing on that interval.

11. Fact
If f is concave up on (a, b), then f (x) ≥ 0 for all x in (a, b)

12. Fact
If f is concave up on (a, b), then f (x) ≥ 0 for all x in (a, b)
If f is concave down on (a, b), then f (x) ≤ 0 for all x in
(a, b).

13. Fact
If f (x) > 0 for all x in (a, b), then f is concave up on (a, b).
If f (x) < 0 for all x in (a, b), then f is concave down on
(a, b).