Understanding Concavity and Inflection Points with Differentiation

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  • 0:06 Review of Concavity
  • 1:46 First Concavity Example
  • 3:06 Second Concavity Example
  • 6:26 Third Concavity Example
  • 9:43 Fourth Concavity Example
  • 11:11 Lesson Summary
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Lesson Transcript
Instructor: Kelly Sjol
Put a little more meaning behind those cups and frowns. In this lesson, use the second derivative of a function to determine if it is concave up or concave down.

Review of Concavity

Let's review concavity for a minute. The concavity of a function is how its derivative is changing, so it's really looking at the second derivative of a function. A function that is concave up looks like a 'cup', and a function that is concave down looks like a 'frown'. If we look at a concave up function, its derivative might be negative or it might be positive, but it's always going up. So the second derivative will always be positive. Something that is concave down has a first derivative that might be positive or negative, but it's always decreasing. So the second derivative of a concave down is going to be negative.

In example #1, the original function is concave up because the second derivative is always positive
Concavity Example 1

If you have something that is concave up and concave down, the point where they meet is called an inflection point. At that point, the second derivative is equal to zero. Here, I have concave up, my smiley face cup, and I've got concave down, my frown. I actually have a minimum and a maximum on this graph, and I have an inflection point where the concave up and concave down parts meet. If I graph the first derivative, I see that it is zero at the minimum and the maximum, and that the second derivative is going to be zero at the inflection point where we go between a cup and a frown.

First Concavity Example

So let's do an example. Is y=x^2 - 1 concave up or concave down? You might be able to graph this off the top of your head, and if you can, you know that it's concave up. It's a parabola that looks like a cup. But how can I actually show that mathematically? Well, is the second derivative positive or negative, or is it both? Let's take the first derivative. I'm going to take the derivative of this with respect to x, which is just y`=2x. Sometimes, when x is less than zero, the first derivative will be negative, and sometimes it'll be positive, when x is greater than zero. Let's take the second derivative, the derivative of y`. So y`` is equal to the derivative of 2x, which is just 2. So the second derivative is always positive, always greater than zero everywhere. Everywhere y=x^2 - 1, our original function, is concave up because the second derivative is always greater than zero.

Second Concavity Example

For the function in example #2, there is an inflection point at x = 1
Concavity Example 2

What about a more complex function like y=(x-1)^3 - 4x+5? I might be able to graph this, but let's formally find out if this is concave up, concave down or both. Let's take the first derivative. So I'm going to differentiate the right-hand side with respect to x and, I have to use the chain rule here, I get y`=3(x-1)^2 - 4. Alright, let's differentiate again: y``=6(x-1). That's both positive and negative. If x is really small, anything less than 1, this is going to become negative. For larger values it's going to be positive. So I know immediately that since the second derivative can be both positive and negative, this graph is going to have regions where I have a cup and regions where I have a frown; it's going to have both. Let's find the inflection points by setting this equal to zero: 0=6(x-1). That's solved when x=1, so at x=1 there's an inflection point. At x=1, I'm going from a frown or a cup to the other one; I'm changing between the two.

Do I have a cup on the left-hand side or a cup on the right-hand side? Let's draw a number line and find out. So I'm going to put x=1 smack dab in the middle of my number line, and I'm going to mark that as my inflection point. I'm going to pick a value on the left-hand side, something less than 1. Zero is usually an easy number so let's pick that point. And I'm going to pick something on the right-hand side of 1. Let's go with 2; I don't want it to be too big. I'm going to see if the second derivative is greater than zero at 0, or is it greater than zero at 2? Maybe it isn't greater than zero at either one. At x=0, my second derivative is going to be y``=6(0-1), or -6. That's less than zero. So for values of x that are less than 1, I have a second derivative that's less than zero, which is going to give me a frown - this concave down region over here. What about at x=2? At x=2, my second derivative is y``=6(2-1), or 6. That's definitely greater than zero. So on the right-hand side, because my second derivative is greater than zero, I have concave up in this region. I have a function that is concave down for x less than 1, and concave up for x greater than 1. At x=1, we have an inflection point.

Third Concavity Example

Graph and number line for the third example
Concavity Example 3

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