 Hi everyone, welcome to this first part of using the second derivative, this part covering how to determine concavity. In the last lesson, we located the intervals in which a function f increases or decreases in order to determine the relative extrema. In this lesson, we will locate the intervals in which f prime, the derivative, increases or decreases in order to determine where the graph of the original function f is curving upward, concave up, or curving downward, concave down. Let f be a differentiable function on an open interval from A to B. We say that the graph of f is concave up if, as you move around the curve, the graph lies above all of its tangent lines, meaning that f prime is increasing on the interval. So let me draw you a picture. So if I were to draw tangent lines here, notice as you go around the curve, the slope is becoming less pronounced. So maybe over here, this top one has a slope of like negative three. I'm just making up estimated numbers. Maybe this second one has a slope of negative two. Maybe this has a slope of negative one. Maybe this has a slope of negative a half. So notice as you come around the curve from the left to the right, those slope values are increasing. Over here, we have positive slopes. And as you go around the curve from the left to the right, they're getting more steep. So maybe down here we have a slope of a half. And maybe this is a slope of one. Maybe this is a slope of two. Maybe this is a slope of three. So once again, as you come around the curve, left to right, those slope values, the derivative are all increasing. The graph of f is said to be concaved down if, as you move around the curve, the graph lies below all of its tangent lines, meaning that the derivative, f prime, is decreasing on the interval. So if we were to once again draw tangent to the curve segments, notice how the tangents all lie on top of the curve. And if we were to try to determine slope values, the derivative values, this one over here, let's start down here at the bottom. That's pretty steep. Maybe that has a slope of like three. Maybe this is a slope of two. Maybe this is a slope of one. They're all positive, but notice as you go around the curve, left to right, those slope values are decreasing. Over here, we have negative slopes. So maybe this is a slope of a negative half. Maybe this is a slope of negative one, perhaps negative two, negative three. And again, they're all negative slopes. But as you come around the curve, those slope values are decreasing. They're getting smaller. Therefore, in accordance with the definition of concavity, in order to find the intervals on which the graph of f is concave up or concave down, you must find the intervals on which f prime is increasing or decreasing. And that brings us to the test for concavity. So suppose function f is differentiable on the open interval from a to b. If the second derivative of f is greater than zero for all x's in the interval from a to b, then the graph of f is concave up. If the second derivative is less than zero for all x's in the open interval from a to b, then the graph of f is said to be concave down. If the second derivative equals zero for all x's in the interval from a to b, then the graph of f is a linear function and is neither concave up nor concave down. So how do you determine, then, where a function is concave up or concave down? Well, if you think about that test for concavity, remember concavity is dictated by the sign, S-I-G-N, of the second derivative. So the first thing we're going to have to do is find the second derivative. Next, we will determine the x values at which the second derivative equals zero or the second derivative does not exist. And we will soon have a name for those x values. Third, we are going to determine the sign of the second derivative on the intervals that are formed by the x values we will have just found. And we are going to do that by way of a number line analysis. Finally, we will then apply the test for concavity. Now, I mentioned that the x values at which the second derivative equals zero or does not exist will soon have a name. We call those points of inflection. If there is a point on the graph of f at which concavity changes from concave up to concave down or vice versa, and if f has only one tangent line there, then the graph will cross the tangent line at this point. This point is called an inflection point. And the sign of the second derivative will change at this point. Let's look at some graphs of some examples for you. So here we have three examples. Notice in the first one on the left, this point right here would be our inflection point. Notice how the curve is concave up. It hits this point, and then it's concave down. And notice the tangent line cuts right through the curve there. In the middle graph, there's your inflection point. Once again, moving from left to right, the curve is concave down. Hits the inflection point, and then is concave up. The third graph, you'll notice that tangent line through the inflection point is vertical. This is one in which the tangent slope obviously is undefined because it's vertical. But once again, we have the curve being concave up first, hitting the inflection point, and then changing to concave down. So by definition, if the point c comma f of c is an inflection point of the graph of f, then either the second derivative equals zero or the second derivative does not exist. It is important to note that the converse of this is not true. Also, just because the second derivative equals zero or the second derivative does not exist, it does not guarantee that you have an inflection point there. You do actually have to go through the number line analysis to find out. So let's recap everything up to this point, and then we'll take and look at an example problem. Critical numbers, remember, occur where either your first derivative equals zero or your first derivative does not exist. Your number line analysis will lead you to information about the intervals on which f is increasing and decreasing, as well as the location of any relative extrema. So the important thing here, critical numbers, you need the first derivative. Possible inflection points, or pips as I like to call them, occur where either the second derivative equals zero or the second derivative does not exist. So once you determine where the second derivative equals zero or does not exist, those x values you get really are only possible inflection points. You need to go through the number line analysis in order to determine if they are definitively inflection points or not. Where you have that change in concavity, that is where you're going to have your inflection point. So let's look at an example problem. And we have a function e raised to the negative x squared over 2. We are asked to determine where it is concave up and concave down and locate the inflection points. So remember, concavity is dictated by the second derivative. So we're going to have to go ahead and find our derivatives. So first derivative will be e raised to the negative x squared over 2. By the chain rule, we need to multiply by the derivative of the exponent. So that's simply going to be negative x. Second derivative will have to do the product rule. So we would have negative e to the negative x squared over 2 plus negative x and e to the negative x squared over 2 times another negative x. And let me simplify that a little bit. So we have negative e to the negative x squared over 2 plus x squared e to the negative x squared over 2. There we go. Now we need to set this equal to 0 and solve for x. And that will give us our possible inflection points, our pips. So this one we could factor out the e term. So we'd have e to the negative x squared over 2. Negative 1 plus x squared. That's really just a difference of two squares. There is no place that this part will equal 0 by applying the 0 product property. So the only possible inflection points we get are positive and negative 1. So if we are going to have any inflection points, and if there's any places that the curve is going to change concavity, these are the candidates. So let's go ahead and do our number line analysis. And hopefully we can come to a conclusion. So the number line analysis is done on the second derivative. Be sure to label your number line. It was at negative 1 and 1 that the second derivative equaled 0. Now we're substituting into the second derivative. So you might need to peek back at what that was. If we substitute something less than negative 1, you should come out with a positive answer. If we substitute something in between negative 1 and positive 1, such as 0, you should obtain a negative answer. And selecting something larger than 1 and substituting into the second derivative, you should get a positive. So from this, we can come to our conclusion about the concavity of the original function f. So let's start with concave up. So our original function f is going to be concave up wherever our second derivative was positive. So that's going to be from negative infinity to negative 1 and 1, 2, infinity. If you go back to your test for concavity, you'll notice that we're all open intervals. Concavity is always, always, always open intervals. Please don't forget f is then concave down only on that middle interval from negative 1 to 1. We have inflection points then at both x equals positive and negative 1. So wherever you have a change in concavity, that is where you have an inflection point.