Inflection Points
Objectives
Students will be able to
• Determine the intervals where a
function is concave up and the intervals
where a function is concave down.
• Find all infection points of a function .
Definition
The point c is called an inflection point
for the function f if there exists an
interval (a, b) about c such that
a. f’’ (x) > 0 in (a, c) and f’’ (x) < 0 in (c, b)
or
b. f’’ (x) < 0 in (a, c) and f’’ (x) > 0 in (c, b)

Test for Inflection Points
Let f be a function with a continuous
second derivative in an interval I,
and let c be an interior point of I.
a. If c is an inflection point for f, then
f’’ (c) = 0.
b. If f’’ (c) = 0 and f’’ changes sign at c,
then c is an inflection point for f.

Example 1
Find the open
intervals where the
function shown in
the graph is concave
upward or concave
downward. Also
indicate any of
inflection points.
Example 2
Find the open
intervals where the
function shown in
the graph is concave
upward or concave
downward. Also
indicate any of
inflection points.
Example 3
Find the open intervals where the function
f (x )  x  12x  45x  2
3
2
is concave upward or concave downward. Find any
of inflection points.
Example 4
Find the open intervals where the function
2
f (x ) 
x 1
is concave upward or concave downward. Find any
of inflection points.
Example 5
Find the open intervals where the function
f (x )  18x  18e
x
is concave upward or concave downward. Find any
of inflection points.
Example 6
The graph to the
right is the graph
of f’ (x), the
derivative of
f(x). Find the
open intervals
where the
function is
concave upward
or concave
downward. Find
any points of
inflection.