Geometry of Functions XI: Inflection Points
Primary tabs
SAMPLE LESSON
Sign in for easy to read lessons narrowly focused on a specific subject. All lessons are linked to each other allowing you to easily see the connections between different Calculus concepts. Our unique Calcmap will give you a bird's-eye view of all our lessons and help you see how important concepts are related to each other. All of the formulas are available to our subscribers, but here is a sample of the lesson you will see when you log in:
In a previous lesson we defined the concept of concavity. A function that looks like a cup is concave up and when it looks like a cup turned upside down it is concave down. We showed that the function <Sign in to see all the formulas> is both concave up and concave down. We instinctively know that the point where this change occurs is at $x=0$. The point where the concavity changes is called an inflection point. This lesson explains how to find inflection points.
The point where a function changes its concavity is called an inflection point. Here is its formal definition:
| Inflection Point |
|---|
| Let $f(x)$ be a function that is continuous at the point $e$. The point $e$ is an inflection point if and only if there exists a <Sign in to see all the formulas> such that $f(x)$ is concave up (or down) on the open interval <Sign in to see all the formulas> and concave down (or up) on the open interval <Sign in to see all the formulas>. |
We admit that the definition of an inflection point is a mouthful. We provide it here because it is a definition you will encounter in more rigorous courses on Calculus. However, you can understand the definition with a simple picture:
If you are looking at the graph and thinking to yourself "I don't get it", then that's OK. It is oftentimes hard to see an inflection point from a function's graph. For example, how could you just look at the graph to see that $e_1$ is an inflection point? Furthermore, our definition of an inflection point is important, but not always very helpful in finding inflection points quickly and easily. What we really need is a way to identify the inflection points easily. Well, here it is:
| If $e$ is an inflection point then either <Sign in to see all the formulas> or <Sign in to see all the formulas> does not exist. |
Be careful in how you use this. You may be tempted to think that finding a point where <Sign in to see all the formulas> or <Sign in to see all the formulas> does not exist means you have found an inflection point. All it means is that you may have found an inflection point. You need to use the formal definition and see if the concavity has changed on either side of your "suspected" inflection point. If it has then you know that it is an inflection point.
Let's look at <Sign in to see all the formulas> as an example of how to use this. We know that $x^3$ is concave down on the open interval <Sign in to see all the formulas> and concave up on the open interval <Sign in to see all the formulas>. From our definition above of an inflection point we know that $x=0$ is an inflection point because it is concave down on the left of $x=0$ and concave up on the right. We already knew a lot about <Sign in to see all the formulas>, but if we didn't we could have looked at the second derivative to get a guess at where the inflection points would be. For example, we know that <Sign in to see all the formulas> and therefore <Sign in to see all the formulas> at $x=0$. Therefore, a guess of the inflection point would be $x=0$. We would then have to show that the concavity changes at that point.
Here is another good example.
