Concavity of Functions for Beginners: Everything You Need to Know
Understanding the concavity of functions for beginners is an important step in learning how the second derivative describes a graph’s behavior. While topics like increasing and decreasing intervals tell us whether a function moves upward or downward, concavity tells us how the graph curves. By studying concavity, we can determine whether a function opens upward, opens downward, or changes its curvature at specific points.
In this guide, we will develop an understanding of concavity. We will define concavity and explain its geometric meaning, learn how to test for concavity using the second derivative, study the second derivative test for classifying extrema, and examine points of inflection in detail. Finally, we will present a step-by-step procedure and work through several examples to demonstrate how these concepts are applied in practice.
By the end of this article, you will have a framework for analyzing the concavity of functions.
What Is Concavity?
When studying functions in calculus, one of our primary goals is to understand the overall shape and behavior of a graph. In the article Secant and Tangent Lines in Calculus: A Complete Beginner’s Guide, we learned that the slope of a tangent line tells us how a function changes at a particular point. In How to Differentiate a Function Step by Step: A Beginner’s Guide, we saw that the derivative provides a way to compute these slopes. Concavity extends these ideas one step further by describing how the slopes themselves change.
Informally, concavity describes the way a graph curves. A function can curve upward or downward. Understanding this curvature helps us visualize functions and identify important features of graphs.
Visual Interpretation of Concavity
Suppose we examine the graph of a function and observe the behavior of its tangent lines. If the slopes of the tangent lines lie below the graph, then the graph curves upward. We say that the function is concave up. In this case, the slope of the tangent lines increases.
On the other hand, if the slopes of the tangent lines lie above the graph, then the graph curves downward. We say that the function is concave down. In this case, the slope of the tangent lines decreases.
Concavity and the First Derivative
Recall that the first derivative \( f'(x) \) measures the slope of the tangent line to the graph of \( f(x) \). Since concavity describes how these slopes change, we can determine concavity by studying the behavior of the first derivative:
- If \( f'(x) \) is increasing, then \( f(x) \) is concave up.
- If \( f'(x) \) is decreasing, then \( f(x) \) is concave down.
This observation naturally leads us to examine the derivative of the derivative or the second derivative.
Concavity and the Second Derivative
The derivative of the first derivative is called the second derivative, denoted by
$$f”(x).$$
The second derivative measures the rate of change of the slope itself, making it the primary tool for analyzing concavity. In general:
- If \( f”(x) > 0 \), then the graph of \( f(x) \) is concave up.
- If \( f”(x) < 0 \), then the graph of \( f(x) \) is concave down.
Thus, while the first derivative tells us whether a function is increasing or decreasing, the second derivative tells us how the function curves. This allows us to understand the overall shape and structure of a graph. In the next section, we will examine how to determine whether a function is concave up or concave down using the second derivative.
Concave Up and Concave Down Functions
Now that we understand that concavity describes how a graph curves, we can develop precise mathematical criteria for determining whether a function is concave up or concave down by examining how the slope of the tangent line changes as we move along the graph.
When Is a Function Concave Up?
A function is said to be concave up on an interval if its graph curves upward. Intuitively, this means that the slopes of the tangent lines are increasing as we move from left to right.
Mathematically, a function \( f(x) \) is concave up on an interval if its second derivative is positive throughout that interval. That is \( f”(x) > 0 \).
Recall that the second derivative measures the rate of change of the first derivative. Therefore, when \( f”(x) > 0 \), the first derivative \( f'(x) \) is increasing.
When Is a Function Concave Down?
A function is said to be concave down on an interval if its graph curves downward. Intuitively, this means that the slopes of the tangent lines are decreasing as we move from left to right.
Mathematically, a function \( f(x) \) is concave down on an interval if its second derivative is negative throughout that interval. That is \( f”(x) < 0 \).
Recall that the second derivative measures the rate of change of the first derivative. Therefore, when \( f”(x) < 0 \), the first derivative \( f'(x) \) is decreasing.
It is important to recognize that concavity is determined by how the slope changes, not by whether the function itself is increasing or decreasing.
This distinction highlights an important principle. The first derivative determines whether a function increases or decreases, and the second derivative determines how the graph curves.
The Second Derivative Test for Classifying Relative Extrema
In the article How to Find Intervals of Increase and Decrease: A Step-by-Step Guide, we learned that relative maxima and minima occur at critical points, which are points where
$$f'(x) = 0$$
or where the first derivative does not exist. We then used sign charts of the first derivative to determine whether the critical points corresponded to local maxima or minima.
Although the first derivative test is a reliable method, a faster approach is often available. When the second derivative exists, we can use the second derivative test to classify critical points without constructing a sign chart for the first derivative.
The second derivative test is based on the relationship between concavity and extrema. If a critical point on a graph is concave up, it must be a local minimum. Conversely, if a critical point occurs on a graph that is concave down, the point must be a local maximum.
More formally, suppose \( c \) is a critical point of \( f(x) \), then
- If \( f”(c) > 0 \), then the graph is concave up at \( x = c \), and has a relative minimum there.
- If \( f”(c) < 0 \), then the graph is concave down at \( x = c \), and has a relative maximum there.
- If \( f”(c) =0 \). The second derivative test is inconclusive.
Why the Second Derivative Test Works
The second derivative test works because concavity determines the local shape of a graph near a critical point. If the graph curves upward, the critical point sits at the bottom of a curve, producing a local minimum. If the graph curves downward, the critical point sits at the top of a curve, producing a local maximum. Thus, the second derivative provides a direct link between the graph’s geometry and the classification of extrema.
Points of Inflection
One application of concavity is the identification of points of inflection. These points mark locations where a graph changes its curvature, transitioning from concave up to concave down or from concave down to concave up.
Definition
A point \( (c, f(c)) \) is a point of inflection if the function changes from concave up to concave down or from concave down to concave up at \( x=c \).
Candidate Points for Inflection
Since points of inflection occur when the concavity changes, the first step is to identify where such changes might occur.
A change in concavity can only occur at points where \( f”(x) = 0 \) or where the second derivative does not exist.
These values are called candidate points of inflection.
How to Verify a Point of Inflection
To determine whether a candidate point is truly a point of inflection, we divide the real line into intervals determined by the candidate points of inflection. Choose test points in each interval, determine the sign of the second derivative on each interval, and check whether the sign changes. If the sign of \( f”(x) \) changes, then the function changes concavity, and the point is a point of inflection.
Step-by-Step Procedure for Determining Concavity and Finding Points of Inflection
By this point, we have developed all the tools needed to analyze the concavity of a function and identify its points of inflection. However, when solving problems, it is helpful to follow a procedure.
Step 1: Compute the First Derivative
Begin by finding the first derivative of the function. Although the second derivative determines concavity, the first derivative is needed because the second derivative is obtained by differentiating \( f'(x) \).
Step 2: Compute the Second Derivative
Next, differentiate the first derivative to obtain the second derivative. The second derivative determines the graph’s concavity.
Step 3: Find Candidate Points of Inflection
To identify where the function may change concavity, solve the equation \( f”(x) = 0 \) and determine where the second derivative is undefined.
Step 4: Divide the Domain into Intervals
Use the candidate points of inflection obtained in Step 3 to divide the domain into separate intervals.
Step 5: Determine Concavity for Each Interval
Choose a test point from each interval and determine the sign of the second derivative. If the sign is positive, the graph is concave up. If the sign is negative, the graph is concave down.
Step 6: Identify and Verify Points of Inflection
A point of inflection occurs only when the sign of the second derivative changes. If the sign of the second derivative does not change, then no point of inflection exists.
Following these steps will allow you to analyze the shape of any function.
Worked Out Example
Example 1: Determine the local maxima, local minima, and points of inflection of \( f(x) = x^3 – 3x^2 – 9x + 5 \).
Solution: Taking the derivative, we get
$$f'(x) = (x^3 – 3x^2 – 9x + 5)’.$$
Using the sum rule, this is
$$f'(x) = (x^3)’ – (3x^2)’ – (9x)’ + (5)’.$$
Taking the derivatives, we get
$$f'(x) = 3x^2 – 6x – 9.$$
Setting the derivative equal to 0, we obtain
$$0 = 3x^2 – 6x – 9.$$
Factoring gives
$$0 = 3(x^2 – 2x – 3).$$
Factoring further, we get
$$0 = 3(x – 3)(x + 1).$$
Dividing both sides of the equation by 3 gives
$$0 = (x – 3)(x + 1).$$
Setting each factor equal to 0, we find
$$x – 3 = 0, x + 1 =0.$$
Adding 3 to both sides of the first equation and subtracting 1 from both sides of the second equation, we obtain
$$x = 3, x = -1.$$
This breaks the real line into the three intervals
$$(-\infty, -1), (-1, 3), (3, \infty).$$
Taking the derivative again, we get
$$f”(x) = (3x^2 – 6x – 9)’.$$
Using the sum rule, this is
$$f”(x) = (3x^2)’ – (6x)’ – (9)’.$$
Taking the derivatives, we get
$$f”(x) = 6x – 6.$$
Setting the second derivative equal to 0, we obtain
$$0 = 6x – 6.$$
Adding 6 to both sides of the equation, we obtain
$$6x = 6.$$
Dividing both sides of the equation by 6 gives
$$x = 1.$$
This breaks the real line into the two intervals
$$(-\infty, 1), (1, \infty).$$
For the first interval, we test
$$-1 \in (-\infty, 1).$$
Substituting this into the second derivative, we get
$$f”(-1) = 6(-1) – 6.$$
Multiplying we find
$$f”(-1) = -6 – 6.$$
Subtracting, we arrive at
$$f”(-1) = -12.$$
For the second interval, we test
$$3 \in (1, \infty).$$
Substituting this into the second derivative, we get
$$f”(3) = 6(3) – 6.$$
Multiplying we find
$$f”(3) = 18 – 6.$$
Subtracting, we arrive at
$$f”(3) = 12.$$
Finally, we conclude that the function is increasing on the intervals \( (-\infty, -1) \) and \( (3, \infty) \), decreasing on the interval \( (-1, 3) \), concave down on the interval \( (-\infty, 1) \) and concave up on the interval \( (1, \infty) \). Furthermore, there is a local minimum at \( x = 3 \), a local maximum at \( x = -1 \), and a point of inflection at \( x = 1 \).
This example demonstrates how the first derivative and second derivative work together. The first derivative identifies and classifies relative extrema, while the second derivative determines the concavity of the graph and locates points of inflection.
Conclusion
Understanding the concavity of functions for beginners is an important step toward developing a deeper understanding of how functions behave. While the first derivative tells us whether a function is increasing or decreasing, the second derivative reveals how the graph curves, allowing us to identify intervals of concavity, classify relative extrema, and locate points of inflection.
In this article, we explored the geometric meaning of concavity, learned how to test for concave up and concave down intervals using the second derivative, studied the second derivative test for classifying local maxima and minima, and examined how to identify and verify points of inflection. We also developed a step-by-step procedure and worked through an example illustrating how these concepts fit together.
Further Reading
How to Find Intervals of Increase and Decrease: A Step-by-Step Guide – The first derivative tells us where a function is increasing or decreasing, providing another way to analyze its behavior.
