*The sign of the second derivative shows which way the graph of a function bends.*

### Definition of Concavity and Concavity Test

Although the sign of the derivative provides information about whether a function is increasing or decreasing, it does not tell us which way the graph of the function bends. The graphs of two increasing functions are shown in Figure 1. The graph on the left bends upward as the point that traces it moves from left to right. In this case, we say the function is concave upward. The graph of the right bends downward, and we say that this function is concave downward.

- Unofficially, we can say a function is concave up on an interval \(I\) if its graph over \(I\) “holds water” and is concave down if the graph “spills water.”

In Figure 2, the graph of the function is concave upward. It is clear from this figure that in passing along the graph from left to right, the slope of the tangent line (which is \(f'(x)\)) increases (before \(P\) the slope is negative, and after \(P\) slope is positive; thus the tangent lines have increasing slopes).

In Figure 3, the graph of the function is concave downward; in passing along the graph from left to right, the slope of the tangent line (or \(f'(x)\)) decreases.

**Definition 1. ** Assume \(f\) is a differentiable function on an interval \(I\).

(a) The (graph of the) function is “concave up” if \(f’\) is increasing on \(I\)

(b) The (graph of the) function is “concave down” if \(f’\) is decreasing on \(I\).

- There are alternative and more general definitions for concavity but the above definition is good enough for this course.

- A line is neither concave up nor concave down.

- Because \(f^{\prime\prime}\) is the derivative of \(f’\), it follows from the Increasing/Decreasing Test that \(f’\) is increasing on an interval \(I\) if \(f^{\prime\prime}(x)>0\) for each \(x\) in \(I\) and \(f’\) is decreasing on \(I\) if \(f^{\prime\prime}(x)<0\) for each \(x\) in \(I\). Therefore, we have the following theorem.

**Theorem 1. Concavity Test.** Let \(f\) be a function whose second derivative exists at each point of an interval \(I\).

(a) If \(f^{\prime\prime}(x)>0\) for every \(x\) in \(I\), then the graph of \(f\) is concave up on \(I\).

(b) If \(f^{\prime\prime}(x)<0\) for every \(x\) in \(I\), then the graph of \(f\) is concave down on \(I\).

### Concavity and Tangent Lines

From Figures 2 and 3, we observe that if a function is concave up on an interval, its graph lies above all of its tangent on that interval and if it is concave down, its graph lies below all of its tangent lines. We can prove this observation is true (see the following example).

### Inflection Points

Most curves are concave up on some interval and concave down on some others. A point where the direction of concavity changes is called an “inflection^{1} point.”

**Definition 2. **We say \((x_{0},f(x_{0}))\) is an inflection point of the graph of \(f\) or simply \(f\) has an inflection point at \(x_{0}\) if:

(a) The graph of \(f\) has a tangent line at \((x_{0},f(x_{0}))\), and

(b) The direction of concavity of \(f\) changes (from upward to downward or from downward to upward) at \(x_{0}\).

- In other words, an inflection point is where the rate of change (the slopes of the tangent lines) changes from increasing to decreasing or from decreasing to increasing.

- We mentioned that if a function is concave up on an interval, its graph lies above the tangents (see Example 3) and if it is concave down, its graph lies below the tangent. If \(P(x_{0},f(x_{0}))\) is an inflection point of the graph of \(f\), the direction of concavity changes and hence the graph crosses one side of the tangent to the other at \(P\).
- Because a tangent line exists at an inflection point, either

\(f'(x_{0})\) must be a finite number, or in the case of a vertical tangent \(f'(x_{0})\) must be \(+\infty\) or \(-\infty\).

### How to Find Inflection Points

**Theorem 2. ** If \((x_{0},f(x_{0}))\) is an inflection point of the graph of \(f\), either \(f^{\prime\prime}(x_{0})=0\) or \(f^{\prime\prime}(x_{0})\) does not exist.

- The above theorem states that if \(f\) has an inflection point at \(x_{0}\), then \(x_{0}\) is a critical point of the derivative \(f’\).
- Even if \(f^{\prime\prime}(x_{0})=0\), the point \((x_{0},f(x_{0}))\) is not necessarily an inflection point. For instance, see the following example.

^{1 }Oxford Dictionary: from Latin inflectere, from in- ‘into’ + flectere ‘to bend’.↗