 Hello, and welcome to the screencaster. We're going to see a couple of examples of finding where a function is concave up and concave down just by looking at its graph. So our first example here is in the blue, and let's just for the sake of argument call this function f. It's a function that is always increasing. And so you can see that f prime is always positive, at least on this interval. But the fact that f prime is positive doesn't really tell the whole story because f is obviously not increasing the same way all the time. And that's what we're trying to get at with the notion of concavity. So let's first of all decide where this function is concave up, and then we'll talk about concave down. A function is concave up, remember, if its first derivative is increasing. The function f here is increasing everywhere, but where is its first derivative is increasing? Another way to think about that is remembering that the first derivative tells you the slope of f. Another way to say this is we're trying to find where the slopes of this graph are increasing. And let's try out just a few sample tangent lines here, starting from the very left edge here. Looks like if I were to draw a tangent line right there would be pretty flat. If I moved over, let's say a couple of units and put a tangent line there, the slope has increased. Okay, I've gone from pretty flat to a little more steep. And if I keep going over to the right for little ways at least, I see that my slopes are getting steeper and steeper and steeper. Up until a certain point, it looks like maybe if I make it to here and draw a tangent line, the slopes are beginning to turn downward a little bit. They're getting less steep. So where are the slopes increasing? On what interval is the first derivative, in other words, increasing? It appears to go from here to, it's kind of hard to say, but maybe about right here, and at this point right here, it looks like the slopes stop increasing and start to decrease. So we'd say on this interval from the first mark there to the fourth mark, it looks as though f prime is not only positive, but f prime is increasing. And that would say that f is concave up there. Other ways to say this are f is increasing at an increasing rate or even it's accelerating upwards. So on that interval there, up to this mystery point right there where I've drawn the big red dot, that is where f is concave up. Now if we continue this process and just keep moving to the right, we'll begin to see that these slopes do continue to drop. And the function gets less and less steep as we move off to the right. In other words, although the function is increasing, the function continues to climb. It's not climbing at an increasing rate anymore. It's climbing at a decreasing rate. And so if I could draw this from say, looks again, looks like it's about here all the way over to the other side. On this side, f prime is still positive, but f prime appears to be decreasing. f is getting less steep, the slopes are decreasing. And so that makes f concave down. So here's an example of a function that is increasing everywhere. But increasing at an increasing rate for a while, and then increasing at a decreasing rate for a while. So here's our second example. It looks more like a graph of say a third degree polynomial. The equation's unimportant. What we're going to try to do here is find where this function, let's call it f again, is concave up or concave down. Now this one's a little different because it is both increasing at some places and decreasing in others. Let's first of all just mark off these intervals where the function f appears to be increasing. That would be from here out to around here. I don't know exactly to say where that turning point is. And then here over to here, that's where your function is increasing. And in the middle is where the function is decreasing. Let's kind of project those points down onto the graph like so, where the turning points are. So let's start deciding about the concavity of this function in the increasing places. So this is going to be a little like the previous example we saw. Over here on the left, the left hand interval where the function is increasing. So f is increasing on this interval. What is f prime doing? Is f prime increasing or is f prime decreasing? The answer to that question is going to tell us concavity. Well, just keep in mind that f prime is the same thing as the slope of f. So if you just imagine some slopes here, you can even draw a few tangent lines on here. If I draw a few tangent lines that go from left to right on this, it's pretty easy to see that they get less steep. The slopes of those tangent lines are decreasing. That means that although f is increasing, f prime is actually decreasing on that interval there. The slopes are getting smaller and that makes f concave down. So concave down on that left-handed interval. Now likewise on the right hand part of this edge of this graph here, if I look at the slopes, it's pretty easy to see that the slopes are getting bigger and bigger and bigger as x moves off to the right. So not only on this right-handed interval is f increasing, the function's heights are getting bigger, its slopes are also getting bigger. So both f and f prime are increasing. And that makes f concave, I'll abbreviate it, concave up on that interval. Now let's examine what happens here in the middle where f is decreasing. We're still asked the same question. The definition of concavity is still the same. Concave up means that the derivative of my function is increasing. So let's look at the slopes of this function as I move from left to right in this middle interval here. Well, let me draw a couple of tangent lines. At the point where the blue dot is, the slope of the tangent line is basically zero, then it drops a little. And then it drops some more, and then eventually kind of evens out again to zero. So the graph starts out with slopes that are zero right there, and then slightly negative, and then slightly more negative. And so that means that the slopes are actually decreasing. If you go from a small negative number to a larger negative number, like negative five and negative ten, that's technically decreasing. And so at least at the outset here from this point forward, the slopes are decreasing. At some point though, and it's hard to say exactly where, but it looks like it's maybe right there on the y-axis. The slopes begin to turn. They go from fairly steep to less steep until you get to this point and they're almost zero again. So we can say that from this point to here, the slopes are decreasing. And so this means that from, let me draw this down underneath, from here to here, my function is concave down, because the slopes are decreasing. F is concave down on that little interval. Likewise, from here to here, the slopes are negative, but they're beginning to be negative and approaching zero. A sequence of negative numbers that approaches zero, just like a temperature indicates something that's increasing. So F prime is increasing on that interval that makes F concave up on that interval. So to put this together, the function is concave down. Actually, I'll put this in black here from here all the way to here, okay? And concave up all the way from here to up to the other edge. So concave down over here, concave up over here. But it's got this interesting mix of concave up and concave down with increasing and decreasing. Thanks for watching.