Using the Second Derivative to Determine Concavity

What Concavity Tells You

The first derivative tells you whether a function is increasing or decreasing. The second derivative tells you how the slope itself is changing. That makes the second derivative the key tool for determining the concavity of a function.

Think about what happens to the slopes as you move from left to right. If the slopes are getting larger, the graph bends upward. If the slopes are getting smaller, the graph bends downward.

So the basic idea is simple:

• Positive second derivative: the slopes are increasing, so the graph is concave up.
• Negative second derivative: the slopes are decreasing, so the graph is concave down.

The Second Derivative Test for Concavity

If is defined on an interval, then:

This is one of the most useful interpretation rules in AP Calculus. You are not simply calculating a derivative—you are using the sign of that derivative to make a statement about the shape of the graph.

A Step-by-Step Method

1. Find the first derivative. You will usually need before you can calculate .
2. Find the second derivative. Differentiate one more time.
3. Find where the second derivative is zero or undefined. These values divide the domain into intervals that may have different concavity.
4. Make a sign chart for the second derivative. Determine whether is positive or negative on each interval.
5. State the intervals of concavity. Positive means concave up; negative means concave down.

Worked Examples

Why the Sign of the Second Derivative Works

The second derivative is the derivative of the first derivative:

That means measures how the slope of the graph is changing.

For example, suppose a graph has slopes of as you move from left to right. The slopes are becoming more positive, so the first derivative is increasing. Therefore, the second derivative is positive and the graph is concave up.

On the other hand, if the slopes change from , the slopes are becoming smaller. The first derivative is decreasing, so the second derivative is negative and the graph is concave down.

Concavity and Inflection Points

An inflection point is a point where the function changes concavity. This is closely related to the second derivative, but there is an important distinction.

A point where is zero is a possible inflection point—not automatically an inflection point.

To confirm an inflection point, the concavity must actually change from up to down or from down to up.

For example, if , then at . But the second derivative does not change sign there, so is not an inflection point.

Common Mistakes

Practice Problems

A Reliable AP Calculus Strategy

When an AP Calculus question asks about concavity, use the same sequence every time:

1. Find .
2. Find .
3. Find where or is undefined.
4. Use those values to divide the domain into intervals.
5. Determine the sign of on each interval.
6. Translate positive into concave up and negative into concave down.
7. If asked for an inflection point, verify that the concavity actually changes.

The key conceptual connection is:

Once you see the second derivative as a measurement of how the slope changes, concavity questions become much more intuitive. You are no longer memorizing a rule—you are interpreting what the derivative is telling you about the graph.