Learning Objectives
A student will be able to:
 Find intervals where a function is concave upward or downward.
 Apply the Second Derivative Test to determine concavity and sketch graphs.
Introduction
In this lesson we will discuss a property about the shapes of graphs called concavity, and introduce a method with which to study this phenomenon, the Second Derivative Test. This method will enable us to identify precisely the intervals where a function is either increasing or decreasing, and also help us to sketch the graph.
 Definition
 A function is said to be concave upward on contained in the domain of if is an increasing function on and concave downward on if is a decreasing function on
 Here is an example that illustrates these properties.
Example 1:
Consider the function :
The function has zeros at and has a relative maximum at and a relative minimum at . Note that the graph appears to be concave down for all intervals in and concave up for all intervals in . Where do you think the concavity of the graph changed from concave down to concave up? If you answered at you would be correct. In general, we wish to identify both the extrema of a function and also the points where the graph changes concavity. The following definition provides a formal characterization of such points.
 Definition
 A point on a graph of a function where the concavity changes is called an inflection point.
 The example above had only one inflection point. But we can easily come up with examples of functions where there is more than one point of inflection.
Example 2:
Consider the function
We can see that the graph has two relative minimums, one relative maximum, and two inflection points (as indicated by arrows).
In general we can use the following two tests for concavity and determining where we have relative maximums, minimums, and inflection points.
Test for Concavity
Suppose that is some interval in the domain of and that is continuous on .
 If for all then the graph of is concave upward on
 If for all then the graph of is concave downward on
A consequence of this concavity test is the following test to identify extreme values of
Second Derivative Test for Extrema
Suppose that is a continuous function near and that is a critical value of Then
 If then has a relative minimum at
 If then has a relative maximum at
 If then the test is inconclusive and may be a point of inflection.
Recall the graph We observed that and that there was neither a maximum nor minimum. The Second Derivative Test cautions us that this may be the case since at at
So now we wish to use all that we have learned from the First and Second Derivative Tests to sketch graphs of functions. The following table provides a summary of the tests and can be a useful guide in sketching graphs.
Signs of first and second derivatives  Information from applying First and Second Derivative Tests  Shape of the graphs 


is increasing is concave upward 


is increasing is concave downward 


is decreasing is concave upward 


is decreasing is concave downward 
Lets’ look at an example where we can use both the First and Second Derivative Tests to find out information that will enable us to sketch the graph.
Example 3:
Let’s examine the function
1. Find the critical values for which
or
at
Note that when
2. Apply the First and Second Derivative Tests to determine extrema and points of inflection.
We can note the signs of and in the intervals partioned by
Key intervals  Shape of graph  

Increasing, concave down  
Decreasing, concave down  
Decreasing, concave up  
Increasing, concave up 
Also note that By the Second Derivative Test we have a relative maximum at or the point
In addition, By the Second Derivative Test we have a relative minimum at or the point Now we can sketch the graph.
Lesson Summary
 We learned to identify intervals where a function is concave upward or downward.
 We applied the First and Second Derivative Tests to determine concavity and sketch graphs.
Multimedia Links
For a video presentation of the second derivative test to determine relative extrema (9.0), see Math Video Tutorials by James Sousa, The Second Derivative Test (8:41).
Review Questions
 Find all extrema using the Second Derivative Test.
 Consider with
 Determine and so that is a critical value of the function
 Is the point a maximum, a minimum or neither?
In problems #3–6, find all extrema and inflection points. Sketch the graph.
 Use your graphing calculator to examine the graph of (Hint: you will need to change the range in the viewing window)
 Discuss the concavity of the graph in the interval .
 Use your calculator to find the minimum value of the function in the interval.
 True or False: has a relative minimum at and a relative maximum at ?
 If possible, provide an example of a nonpolynomial function that has exactly one relative minimum.
 If possible, provide an example of a nonpolynomial function that is concave downward everywhere in its domain.
Review Answers
 There is a relative minimum at the relative minimum is located at
 suggests that and solving this system we have that ; the point is an absolute minimum of
 Relative maximum at , relative minimum at ; the relative maximum is located at ; the relative minimum is located at There is a point of inflection at .
 Relative maximum at , located at ; relative minimum at , located at . There are no inflection points.
 Relative maximum at relative minimum at ; the relative maximum is located at ; the relative minimum is located at There is a point of inflection at
 Relative maximums at relative minimum at ; the relative maximums are located at and ; the relative minimum is located at There are two inflection points, located at and
 The graph is concave up in the interval;
 There is a relative minimum at
 False: there are inflection points at and . There is a relative minimum at
 on Also,