In this section we learn about the two types of curvature and determine the curvature of a function.
Concavity
In this section we will discuss the curvature of the graph of a given function. There are two types of curvature: concave up and concave down. The main tool for discussing curvature is the second derivative, .
Concavity Suppose is differentiable on an open interval, . If is increasing on , then
is concave up on and if is decreasing on , then is concave down on
.
The following theorem helps us to determine where a function is concave up and
where it is concave down.
Concavity If on an interval , then is concave up on and if on an interval , then is
concave down on .
example 1 Find intervals where the function is concave up or concave down.
The second derivative is given by Since for all values of , is concave up on the interval . This is clear based on the graph of as well.
The second derivative is given by Since for all values of , is concave up on the interval . This is clear based on the graph of as well.
example 2 Find intervals where the function is concave up or concave down.
The second derivative is given by Since for all values of , is concave up on the interval . This is clear based on the graph of as well.
The second derivative is given by Since for all values of , is concave up on the interval . This is clear based on the graph of as well.
example 3 Find intervals where the function is concave up or concave down.
The second derivative is given by Since has the same sign as itself, is concave up on the interval and concave down on the interval . This is clear based on the graph of as well.
The second derivative is given by Since has the same sign as itself, is concave up on the interval and concave down on the interval . This is clear based on the graph of as well.
The last example brings up a new concept. The function changes concavity at . We
call this an inflection point of the function.
In the next two examples, we will discuss the curvature of the given functions and find their inflection points.
example 4 Determine where the cubic polynomial is concave up, concave down and
find the inflection points.
The second derivative of is . To determine where is positive and where it is negative, we will first determine where it is zero. Hence, we will solve the equation for . 
The second derivative of is . To determine where is positive and where it is negative, we will first determine where it is zero. Hence, we will solve the equation for .
We have so . This value breaks the real number line into two intervals, and . The second derivative maintains the same sign throughout each of these intervals. To determine whether it is positive or negative, we choose a test point in each interval. For the interval, , we choose . Plugging this into the second derivative, we get . Next, we choose the test point for the interval . Plugging this into the second derivative gives . We can use this information to create a sign chart for the second derivative, as shown below.
We can now use the concavity theorem to conclude that the original function, is concave down on the interval . and concave up on the interval . Finally, noting that the original polynomial is continuous everywhere and changes concavity at , we can say that has an inflection point at .
example 5 Determine where the cubic polynomial is concave up, concave down and
find the inflection points.
The second derivative of is To determine where is positive and where it is negative, we will first determine where it is zero. Hence, we will solve the equation for . 
The second derivative of is To determine where is positive and where it is negative, we will first determine where it is zero. Hence, we will solve the equation for .
We have so . This value breaks the real number line into two intervals, and . The second derivative maintains the same sign throughout each of these intervals. To determine whether it is positive or negative, we choose a test point in each interval. For the interval, , we choose . Plugging this into the second derivative, we get .
Next, for the interval , we choose the test point . Plugging this into the second derivative gives .
We can now use the concavity theorem to conclude that the original function, is concave down on the interval , and concave up on the interval . Finally, noting that the original polynomial is continuous everywhere and changes concavity at , we can say that has an inflection point at .
(problem 5a) Find the inflection point(s) of the function
List multiple answers in ascending order.
The function has inflection points at and .
(problem 5b) Find the inflection point(s) of the function
If there are no inflection points, type “none”.
The function has inflection point(s) at
The function has inflection point(s) at
(problem 5c) Find the intervals of increase/decrease, local extremes, intervals of
concavity and inflection points for the function
example 6 Determine where the function is concave up, concave down and find the
inflection points.
To find , we will need to use the product rule twice. First, and second Now that we have the second derivative, we set it equal to zero. Solve for . Since the exponential is never equal to zero, the only solutions come from setting the quadratic to zero: This quadratic does not factor, so we need to use the quadratic formula. The solutions are For simplicity, we will call these two roots and . So and 
To find , we will need to use the product rule twice. First, and second Now that we have the second derivative, we set it equal to zero. Solve for . Since the exponential is never equal to zero, the only solutions come from setting the quadratic to zero: This quadratic does not factor, so we need to use the quadratic formula. The solutions are For simplicity, we will call these two roots and . So and
These two values break the real number line into three intervals: and with test points and , respectively. Plugging these into the second derivative gives and
We can now use the concavity theorem to conclude that the original function, is concave up on the intervals and and it is concave down on the interval . Finally, we can infer from this that the continuous function has inflection points at
(problem 6b) Find the inflection point(s) of the function
List multiple answers in ascending order.
The function has inflection points at and
The function has inflection points at and
(problem 6c) Find the inflection point(s) of the function in the interval
List multiple answers in ascending order.
On the interval , the function has inflection points at and .
List multiple answers in ascending order.
On the interval , the function has inflection points at and .
example 7 Determine where the function is concave up, concave down and find the
inflection points.
By the chain rule, its derivative is and by the product rule (and the chain rule again, as well), its second derivative is Setting the second derivative equal to zero and noting that the exponential is always positive, we get Solving for gives 
By the chain rule, its derivative is and by the product rule (and the chain rule again, as well), its second derivative is Setting the second derivative equal to zero and noting that the exponential is always positive, we get Solving for gives
These two values of break the real number line into three intervals: and . The sign of will remain the same on each of these intervals. To determine the sign on each interval, we use the test points and respectively. Plugging the test points into the second derivative gives and
We can now use the concavity theorem to conclude that the original function, is concave up on the intervals and and concave down on the interval . Finally, since the original function is continuous everywhere, we can say that are inflection points for .
(problem 7) Find the inflection point(s) of the function
List multiple answers in ascending order.
The function has inflection points at and .
The function has inflection points at and .
example 8 Determine where the cubic polynomial is concave up, concave down and
find the inflection points.
From the power rule with , we have and its second derivative is (power rule, ) Now we make the observation that has the same sign as and so will have the same sign as in this example. Thus on the interval and on the interval We conclude that is concave up on and concave down on the interval Since is not continuous at (it has an infinite discontinuity there), there is no inflection point there.
From the power rule with , we have and its second derivative is (power rule, ) Now we make the observation that has the same sign as and so will have the same sign as in this example. Thus on the interval and on the interval We conclude that is concave up on and concave down on the interval Since is not continuous at (it has an infinite discontinuity there), there is no inflection point there.
(problem 8) Find the inflection point(s) of the function on the interval .
On the interval , the function
has an inflection point at .
On the interval , the function
has an inflection point at .
Using the graph of the derivative
In this section, we are given the graph of the derivative, , and we are asked to make conclusions about the original function, .
example 9 Use the graph of , shown below, to answer the following questions about
the graph of . Where is increasing/decreasing and where are its local extremes?
Where is concave up/down and where are its inflection points?
To answer the first question, recall that when , then is increasing, and when ,
then is decreasing. From the graph, we see that is positive on the intervals
and . Hence, the graph of is increasing on these intervals. We can also
see that is negative on the interval and therefore is decreasing on this
interval. We can now use the first derivative test to determine the nature of
the local extremes. Since and changes sign from positive to negative at
, the function, , has a local maximum at . Similarly, Since and changes
sign from negative to positive at , the function, , has a local minimum at
.
To determine the concavity of ,recall that is concave up when is increasing and is
concave down when is decreasing. From the graph, we see that is increasing on the
interval , and decreasing on the interval . Hence, the graph of is concave up on and
concave down on . Finally, has an inflection point at due to the change in concavity
there.
(problem 9) Use the graph of the derivative, , given below, to answer the following
questions.
Where is increasing? Select all that apply.
Where is decreasing? Select all that apply.
Describe the local extremes of . Select all that apply.
Where is concave up? Select all that apply.
Where is concave down? Select all that apply.
Where are the inflection points of ? Select all that apply.
example 10 Use the graph of , shown below, to answer the following questions about
the graph of . Where is increasing/decreasing and where are its local extremes?
Where is concave up/down and where are its inflection points?
To answer the first question, recall that when , then is increasing, and when ,
then is decreasing. From the graph, we see that is positive on the intervals
and . Hence, the graph of is increasing on these intervals. We can also
see that is negative on the interval and therefore is decreasing on this
interval. We can now use the first derivative test to determine the nature
of the local extremes. Since and changes sign from negative to positive
at , the function, , has a local minimum at . The situation at is different.
From the graph, we see that but does not change sign at - it is positive on
either side. Thus, does not have a local extreme at the critical number,
.
To determine the concavity of ,recall that is concave up when is increasing and is
concave down when is decreasing. From the graph, we see that is increasing on the
intervals and and decreasing on the interval . Hence, the graph of is concave up on
and and concave down on . Finally, has an inflection points at and due to the
changes in concavity at these points.
(problem 10) Use the graph of the derivative, , given below, to answer the following
questions.
Where is increasing? Select all that apply.
Where is decreasing? Select all that apply.
Describe the local extremes of . Select all that apply.
Where is concave up? Select all that apply.
Where is concave down? Select all that apply.
Where are the inflection points of ? Select all that apply.
Video lessons
Here is a detailed, lecture style video on concavity:
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We next explore a special relationship between concavity and local extremes.
Second Derivative Test Suppose . If , then has a local maximum at , and if , then
has a local minimum at
The following figure should convince the reader of the validity of the Second Derivative Test.
