This document discusses techniques for analyzing functions based on their derivatives. It explains how to determine if a function is increasing or decreasing based on the sign of the first derivative. It also describes how to identify relative maxima and minima by examining changes in the sign of the first derivative. Additionally, it covers how to determine if a function is concave up or down using the second derivative and how to identify points of inflection where the concavity changes. The second derivative test is presented for classifying critical points as maxima or minima.

1. Sections
3.3-3.4
First and Second
Derivative Information

2. Test for Increasing or Decreasing
Functions
Let f be continuous on [a,b] and differentiable on (a,b).
1. If ( ) 0 for all x in (a,b), then f is increasing on [a,b].
2. If ( ) 0 for all x in (a,b), then f is decreasing on [a,b].
3. If ( ) 0 for all x in (a,b), then f is constant on [a,b].
f x
f x
f x
′ >
′ <
′ =

3. Increasing/Decreasing
To determine whether the function is increasing or
decreasing on an interval, evaluate points to the left and right
of the critical points on an f’ numberline.
f '(x)
c
+ __
inc
__dec
f '(x)
c
+__
incdec

4. First Derivative Test
Let c be a critical number of a function f that is
continuous on an open interval containing c. If f is
differentiable on the interval, except possibly at c, then
f(c) can be classified as follows…
1. If ( ) changes from negative to positive at c,
then f(c) is a relative minimum.
2. If ( ) changes from positive to negative at c,
then f(c) is a relative maximum.
f x
f x
′
′

5. 1st Derivative Test
f '(x)
c
+
inc
__dec
If the sign changes from + to - at c, then c is a relative maximum.
Max
f '(x)
c
+__ incdec
If the sign changes from - to + at c, then c is a relative minimum.
Min

6. • A curve is concave up if its slope is increasing, in which case
the second derivative will be positive ( f "(x) > 0 ).
• Also, the graph lies above its tangent lines.
•A curve is concave down if its slope is decreasing, in which
case the second derivative will be negative (f "(x) < 0 ).
• Also, the graph lies below its tangent lines.
Concavity

7. Test for Concavity
Let f be a function whose 2nd
derivative exists on
an open interval I.
1. If ( ) 0 for all x in I, then f is concave upward.
2. If ( ) 0 for all x in I, then f is concave downward.
f x
f x
′′ >
′′ <

8. Concavity Test
To determine whether a function is concave up or concave
down on an interval, determine where f "(x) = 0 and f "(x) is
undefined. Then evaluate values to the left and right of these
points on an f " numberline.
f "(x)
c
+ __
ccu
__ccd
f "(x)
c
+__
ccuccd

9. A point where the graph of f changes concavity, from
concave up to concave down or vice versa, is called a
point of inflection. At a point of inflection the second
derivative will either be undefined or 0.
Inflection

10. If the signs on the f "(x) numberline do not change, then
c is not an inflection point.
f "(x)
c
__
ccdccd __
Not inflection
When the signs change on an f " numberline, there is an
inflection point.

11. Second Derivative Test
Let f be a function such that f’(c) = 0 and the 2nd
derivative of f exists on an open interval containing c.
1. If ( ) 0 , then f(c) is a relative minimum.
2. If ( ) 0,then f(c) is a relative maximum.
3. If ( ) 0, then the test fails. Use the 1st Derivative Test.
f c
f c
f c
′′ >
′′ <
′′ =