lec

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
′′ >
′′ <
′′ =