 Hi there, and welcome to another screencast, and in this one we're going to talk about inflection points. And the goal of this screencast is that we're going to take a graph of a function, and just by looking at it, just by visual means, identify all the inflection points on the graph of this function. Now before we do that, we're going to have a little review here about concavity. Just remember that the definition of an inflection point is, it's a point on the graph of a function where the function's concavity changes from concave up to concave down, or vice versa. So we need to understand concavity and literally what concavity looks like before we have a chance of accomplishing the goal of the screencast. So I have a little grid here that we're going to draw little caricatures of all the possible combinations of concave up, concave down, and linear on one hand versus increasing and decreasing on the other hand. So first of all, what does an increasing concave up function look like? Well that will be a function that's increasing. So as I move to the right, the height of the function goes up, but also increasing at an increasing rate. So that sort of function looks like this. It's going up and sort of bending upward as we move upward. Now on the other hand, what does an increasing concave down function look like? Well that will be a function that of course is still increasing, but at a decreasing rate as I increase. So increasing at a decreasing rate looks like this. It's going up, but it's bending downward as it goes up. Almost looks like it's leveling off, although there may not be an actual leveling off taking place. Now an increasing linear function, one that's increasing but doesn't have any concavity, it all would be just a straight line that's going up. Now let's think about decreasing and these different combinations of concavity. A function of this decreasing and concave up is a function that's going down, moving downward as I move to the right, but bending upward. That function would look like this. Decreasing but bending upward. A decreasing concave down function is decreasing but bending downward. That looks like this. And of course decreasing linear is just a straight line that's going down. So especially pay attention to these four little pieces of the grid right here. Every function that is differentiable at all and that is not linear looks like some combination of those four pieces. And it's where those two of those four pieces come together is where your inflection points are going to be. So now let's go to a sample graph and see if we can apply this idea to finding the inflection points. So here's a graph of a function. We don't have a formula for it. We don't even have skate marks on the x and y axes. But I'd like to find all the inflection points on the graph of this function. First of all I'd like you to have a chance at this. So look at this function carefully for a minute or two and see if you can identify all the inflection points that are on it and then write down how many inflection points do you think this function has. So pause the video and look at the graph for a minute. Write down your answer and then come back and we'll talk about it. So the answer is that this function has five inflection points. Let's see if we can find them all. Now we're going to kind of work backwards here by first of all identifying all the places where the graph is concave up and concave down. I'm going to do that by drawing on top of the graph here. I'm going to draw all the places where my function is concave up in green and all the places where the function is concave down in red. So let me switch over to red here because we're going to start the graph over here at the far left. And if you think back to the pictures of concave up and concave down that we just drew from here up to around here the function is increasing and concave down. So I'm going to mark that with red because that's concave down so far. Then I hit a maximum point but that's not an inflection point because from here to right around here the function is still concave down. It's decreasing concave down. So I'm going from increasing concave down to decreasing concave down. That would make this point right here where the function changes from increasing to decreasing a critical value but it's not an inflection point because the concavity didn't change. Now on the other hand at this point where I stopped from here to around here this looks like decreasing concave up. It looks like that piece of the graph that we drew a few minutes ago where I'm decreasing but I'm bending upward. And so one inflection point that I see is right there at the junction between the decreasing concave down and decreasing concave up. That's one of the inflection points. Let's keep drawing and see if we can identify the other four. So once I pass this local minimum value right here where my green stops I'm going to start climbing but just because I start climbing doesn't mean I have an inflection point. I have a critical value but not an inflection point. I'm going to continue to climb and it sort of looks like the increasing concave up graph and in fact that's what it is. Until right around here and at this point looks like we switch from increasing concave up to increasing concave down until right here. That definitely that red segment I just drew definitely looks like the increasing but bending downward situation. So right there at this junction point is a second inflection point. At this point my graph changes from increasing concave down to increasing concave up for a while and that this piece sort of looks like this piece that we drew a while ago. So here's a third inflection point. Now let's see if we can continue here. At this point where the green stops I'm changing over from increasing concave up to increasing concave down again for just a little bit. That's an increasing concave down piece. So that junction point is a fourth inflection point. Now where my red stops here I'm changing from increasing to decreasing. That's a local maximum value there but I'm not changing concavity. I'm changing from increasing concave down to decreasing concave down until somewhere around here. That piece kind of looks like this piece we started with over here. So I have not encountered an inflection point yet but I think I'm going to hit another one right where I stop because I'm going from not changing from decreasing to increasing but decreasing concave down to decreasing concave up like so and so that is a fifth inflection point. And at the very end of this graph I am changing from decreasing concave up to increasing concave up but that doesn't give me any new inflection points. So those five places one two three four five where the concavity changes from down to up or vice versa those are my five inflection points. So to sum this all up if you want to find the inflection points on the graph of a function what you really have to do is identify the segments of the function that are concave up and concave down. You can do that visually by labeling them whatever works for you but it's where those concave up regions change to concave down regions and vice versa that's where the inflection points are. Thanks for watching.