1) The document discusses tests for determining whether a function is increasing, decreasing, or constant based on the sign of the first derivative on an interval. 2) It also describes how to use the first and second derivative tests to determine if a critical point is a relative maximum or minimum. The tests analyze how the sign of the first or second derivative changes at the critical point. 3) Finally, the document outlines the test for concavity and points of inflection based on the sign of the second derivative on an interval. A change in the sign of the second derivative indicates an inflection point.

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 f ′( x) > 0 for all x in (a,b), then f is increasing on [a,b].
2. If f ′( x) < 0 for all x in (a,b), then f is decreasing on [a,b].
3. If f ′( x) = 0 for all x in (a,b), then f is constant on [a,b].

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.
inc
f '(x)
dec
__
+
c
dec
inc
__
+
f '(x)
c

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 f ′( x) changes from negative to positive at c,
then f(c) is a relative minimum.
2. If f ′( x) changes from positive to negative at c,
then f(c) is a relative maximum.

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

6. Concavity
• 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.

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

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.
ccu
ccd
f "(x)
__
+
c
ccd
ccu
__
+
f "(x)
c

9. Inflection
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.

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

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 f ′′(c) > 0 , then f(c) is a relative minimum.
2. If f ′′(c) < 0, then f(c) is a relative maximum.
3. If f ′′(c) = 0, then the test fails. Use the 1st Derivative Test.