 So here's another useful graph property that translates into some information about the derivative. So let's consider the graph of some function y equals f of x, and let's say it looks something like this. And one thing we might notice here is that if I draw the tangent lines to this graph, they're all below the graph. So here's a tangent line, the tangent line is below the graph, another tangent line below the graph, another tangent line below the graph. Everywhere that I draw a tangent line, the graph itself is going to be above the tangent line, the tangent line is going to be below the graph. And in that case, we say that the graph is concave upwards. Well, if we have a concave upwards, then we must have a concave downwards. And that occurs if I have a different graph, something like this, where in this case, if I look at the tangent lines, they're all above the graph. So again, here I have my tangent line is above the graph, my tangent line is above the graph, my tangent line is above the graph. And in this case, we have a graph that is concave downwards. And the universe is rarely black and white. It's often filled with shades of gray, not 50 of them. But I might consider a graph maybe something like this. And here the thing to notice is that sometimes I have parts of the graph that are concave down. So if I draw the tangent line, the graph is actually below the tangent line. And there's other sections of the graph where they are concave up. Where if I draw the tangent line, the curve is actually above the graph. And so what happens is my concavity changes from, in this case, down to up. And the place where that concavity change occurs is called an inflection point. And in this case, it's around here someplace. Before this point, the tangent lines are all above the graph. After this point, the tangent lines are all below the graph. And around here, that behavior changes. So let's consider some graph and suppose the second derivative does actually exist. In general, we have the following information. If the second derivative is positive at some point, the graph and the vicinity of that point is going to be concave up. And likewise, if the second derivative is negative at a point, the graph and the vicinity of that point is going to be concave down. What this means is that if I have any hope for an inflection point, it has to be where the graph changes concavity from up to down, down to up. In other words, second derivative positive to second derivative negative, or vice versa. Which means that every point of inflection corresponds to a place where the second derivative changes sign. And if I can locate those points where the second derivative changes sign, I know where the inflection points are. So let's take a look at that. So I have a function, I want to describe the concavity, and so on. So let's find that second derivative. So our function is x cubed minus 3x squared plus 12x minus 125. Standard polynomial differentiating is not too complex. And so my second derivative looks like 6x minus 6. Now the important thing to remember here is that I want to find where the second derivative is positive or negative. And that means that I'm going to have to look for where the second derivative is either zero or it fails to exist. Now here's an important note on how we write mathematics. There's a great temptation to say, well I want to find second derivative equal to zero. So I'll say second derivative is this equal to zero. If you write that, it is misleading, which is even worse than being wrong. If you write that, it suggests that second derivative is zero. Again, the two things on any side when equals have a guarantee of being completely and totally interchangeable. So if you write second derivative 6x minus 6 equals zero, you have just guaranteed the second derivative is zero. It's not. It's 6x minus 6. What you actually want to write is on a separate line, the second derivative expression, not the second derivative symbol itself, but what the expression is, and find when that's equal to zero. And what this says is, well we're actually solving an equation here. I want to solve the equation 6x minus 6. And after a lot of algebra, I find that x is equal to 1. And so what does that tell me? Well again, if I'm looking for the inflection points, I need to know where the second derivative is positive, second derivative is negative. Now, x equals 1 is where second derivative is zero. That tells me absolutely nothing about whether the second derivative is positive or negative any place else, except whatever it is, it can't change, except by going through this point. So I want to find where the second derivative is positive. Well, if x is greater than 1, my second derivative is going to be positive, so the graph of y equals f of x will be concave up. Likewise, if x is less than 1, the second derivative is negative, so the graph of y equals f of x is going to be concave down, and here's the important thing, there is a change in concavity at x equals 1. We go from being concave down to being concave up, there is a change in concavity, and so that means that we have a point of inflection or an inflection point. One last thing to notice, even though inflection points and concavity are all about properties of a graph, we solve this problem completely without having to graph y equals f of x, and what that leads us to is I can find a lot of information about what a graph looks like before I even try to graph it, and that leads us to a major topic. How do we graph using the derivative? We'll take a look at that later.