# Convex and Concave functions and inflection points

**Convex** and **concave** are words that we use to describe the shape or curvature of a curve. Recall from classifying stationary points (see Stationary Points page) that we can find the second derivative of a function by differentiating twice. We also require the second derivative to see if a function is concave or convex at a particular point. A **point of inflection** is where a curve changes from being concave to being convex or vice versa.

## Convex and Concave Functions

Recall firstly that solving for will identify the locations of the stationary points of the curve . Secondly, finding the value of at these stationary points allows us to classify the stationary points. That is if , it is a maximum and if , it is a minimum.

It is, however, possible to find the value of at any point on a curve, not necessarily stationary points. If at any point on a curve, the second derivative is negative, we say that the curve is **concave** at that point. Conversely, if the second derivative is positive at any point, we say that the curve is **convex** at that point. It follows that there is an interval around a maximum that is concave and an interval around a minimum that is convex. See Example 1. The point where a curve changes from being concave to convex or vice versa is known as an inflection point.

It is, however, possible to find the value of at any point on a curve, not necessarily stationary points. If at any point on a curve, the second derivative is negative, we say that the curve is **concave** at that point. Conversely, if the second derivative is positive at any point, we say that the curve is **convex** at that point. It follows that there is an interval around a maximum that is concave and an interval around a minimum that is convex. See Example 1. The point where a curve changes from being concave to convex or vice versa is known as an inflection point.

Consider the graph of . The derivative is and so the second derivative is . This is always positive and so the curve is always convex. Similarly for the graph of , and so this curve is always concave. It can help to remember that a concave curve has the shape of an actual cave.

## Points of Inflection

Consider the graph of . The derivative is and the second derivative is . The second deriative is 0 when , it is positive when and negative when . It follows that the point is an inflection point. Also, the curve is concave when and convex when . A point of inflection is where a curve goes from being concave to convex or vice versa. This means that the second derivative changes sign. For the curve of the point is a stationary point as well as a point of inflection. However, an inflection point doesn’t have to be a stationary point as we can see in the section above. Also see Example 2. It is the fact that and changing sign that makes a given point one of inflection.