Transcript 3 4 Concavity 2nd Derv Test

3-4: Concavity and the Second Derivative Test
Objectives:
Assignment:
1. To determine the
concavity and points
of inflection of the
graph of a function
• P. 195-197: 1-25 odd,
49-52, 76, 79-82
2. To apply the Second
Derivative Test to
find relative extrema
• P. 195: 17-39 odd
Concave Upward
Informally, a
graph is
concave
upward if it
curves up.
Concave Downward
Informally,
a graph is
concave
downward
if it curves
down.
Warm-Up
Curve your pipe cleaner to fit the following
descriptions:
𝑓 is increasing
and concave
upward
𝑓 is decreasing
and concave
upward
𝑓 is increasing
and concave
downward
𝑓 is decreasing
and concave
downward
Warm-Up
In each case, where does the curve appear
in relation to its tangent lines?
𝑓 is increasing
and concave
upward
𝑓 is decreasing
and concave
upward
𝑓 is increasing
and concave
downward
𝑓 is decreasing
and concave
downward
Warm-Up
Curve your pipe cleaner to fit the following
descriptions:
𝑓 is increasing
at an
increasing rate
𝑓 is decreasing
at a decreasing
rate
𝑓 is increasing at
a decreasing
rate
𝑓 is decreasing
at an
increasing rate
Concavity
is related
to the rate
of change
of the rate
of change.
Objective 1
You will be able
concavity and
of the graph
to determine the
points of inflection
of a function
Exercise 1
Notice that the graph
of 𝑓(𝑥) = 𝑥 2 − 4𝑥 + 3
is concave upward.
What is true about
the derivative of 𝑓?
What is true about
the second derivative
of 𝑓?
Definition of Concavity
Let 𝑓 be differentiable on an open interval 𝐼.
The graph of 𝑓 lies
above all of its
tangent lines on 𝐼.
The graph of 𝑓 is
concave upward on
𝐼 if 𝑓′ is increasing
on the interval.
Definition of Concavity
Let 𝑓 be differentiable on an open interval 𝐼.
The graph of 𝑓 is
concave downward
on 𝐼 if 𝑓′ is decreasing
on the interval.
The graph of 𝑓 lies
below all of its
tangent lines on 𝐼.
Test for Concavity
Let 𝑓 be a function whose second derivative
exists on an open interval 𝐼.
If 𝑓′′(𝑥) > 0
for all 𝑥 in 𝐼,
then the
graph of 𝑓 is
concave
upward in 𝐼.
If 𝑓′′(𝑥) < 0
for all 𝑥 in 𝐼,
then the
graph of 𝑓 is
concave
downward in
𝐼.
If 𝑓′′(𝑥) = 0
for all 𝑥 in 𝐼,
then 𝑓 is
linear and
has no
concavity.
Test for Concavity
Let 𝑓 be a function whose second derivative
exists on an open interval 𝐼.
If 𝑓′′(𝑥) < 0
If 𝑓′′(𝑥) > 0
If 𝑓′′(𝑥) = 0
for all 𝑥 in 𝐼,
for all 𝑥is
in 𝐼,
for all 𝑥𝑓inis
𝐼,
This
similar tothen
testing
for where
the
then the
then 𝑓 is
graph
of
𝑓
is
increasing
or decreasing, except
graph
of 𝑓 is
linear and
concavend
concave
has no
you’re using
the 2 in derivative.
downward
upward in 𝐼.
concavity.
𝐼.
Exercise 2
Determine the open intervals which the
6
graph of 𝑓 𝑥 = 2 is concave upward or
𝑥 +3
downward.
Exercise 3
Determine the open intervals on which the
𝑥 2 +1
𝑥 2 −4
graph of 𝑓 𝑥 =
is concave upward or
concave downward.
Points of Inflection
Let 𝑓 be a function that is continuous
on an open interval and let 𝑐 be a
point in the interval.
If the graph of 𝑓 has a tangent line
at this point 𝑐, 𝑓 𝑐 , then this
point is a point of inflection if the
concavity of 𝑓 changes from
upward to downward (or downward
to upward) at the point.
Points of Inflection
Let 𝑓 be a function that is continuous
on an open interval and let 𝑐 be a
point in the interval.
If 𝑐, 𝑓 𝑐 is a point of inflection
of the graph of 𝑓, then either
𝑓′′(𝑐) = 0 or 𝑓′′ does not exist at
𝑥 = 𝑐.
Exercise 4
Determine the points of
inflection and discuss the
concavity of the graph of
𝑓(𝑥) = 𝑥 4 .
Exercise 5
Determine the points of inflection
and discuss the concavity of the
graph of 𝑓(𝑥) = 𝑥 4 − 4𝑥 3 .
You will be able to
apply the Second
Derivative Test to
find relative
extrema
Objective 2
Exercise 6a
Note that
1 3
𝑓 𝑥 = 𝑥 − 2𝑥 2 + 3𝑥 + 1
3
has a relative maximum
at 𝑥 = 1.
What is the value of the
second derivative at 𝑥 = 1?
Exercise 6b
Note that
1 3
𝑓 𝑥 = 𝑥 − 2𝑥 2 + 3𝑥 + 1
3
has a relative minimum at
𝑥 = 3.
What is the value of the
second derivative at 𝑥 = 3?
Second Derivative Test
Let 𝑓 be a function such that 𝑓′(𝑐) = 0 and
the second derivative of 𝑓 exists on an open
interval containing 𝑐.
If 𝑓′′(𝑐) > 0, then
𝑓 has a relative
minimum at
𝑐, 𝑓 𝑐 .
Second Derivative Test
Let 𝑓 be a function such that 𝑓′(𝑐) = 0 and
the second derivative of 𝑓 exists on an open
interval containing 𝑐.
If 𝑓 ′′ 𝑐 < 0, then
𝑓 has a relative
maximum at
𝑐, 𝑓 𝑐 .
Exercise 7
Find the relative extrema for
𝑓(𝑥) = −3𝑥 5 + 5𝑥 3 .
Possible Max
Possible Min
𝑓 𝑥
Increasing
Critical Points
Decreasing
Concave
Upward
𝑓′ 𝑥
+
0 or undefined
−
Increasing
𝑓′′ 𝑥
+
−
+
Possible
Points of
Inflection
Concave
Downward
Decreasing
0 or
undefined
−
3-4: Concavity and the Second Derivative Test
Objectives:
Assignment:
1. To determine the
concavity and points
of inflection of the
graph of a function
• P. 195-197: 1-25 odd,
49-52, 76, 79-82
2. To apply the Second
Derivative Test to
find relative extrema
• P. 195: 17-39 odd