Hi and welcome to the module, the last module of videos that we're going to have in this course of this specialization. We're going to talk today about maxima and minima of our function. So, it means the maximum is the highest point. The minimum is the lowest point. And you know that we've talked about how where the derivative where the slope becomes 0 at the top or the bottom. Right, that's the top of the bottom where it sort of turns around. So we could look at a function that looks like this. Let's just see this one is the graph of f of x. Oops, I got it right where you can see it. F of x equals minus x squared plus x plus 3. Okay, so for this graph of this function, it's going to be increasing and then decreasing. Okay, so we can see now I got the whole function on the page now. You can see the function is increasing and then it's decreasing. And so it will have what we call an absolute maximum. The highest point on the entire function. In other words, no place does the function ever get higher than this. And then what happens there is the derivative is 0. So, this would be an example of an absolute maximum. Now, if the thing was turned upside down say something like that, which could be the graph of just the negative of this function. That would be x squared minus x minus 3. Okay, it's going to have an absolute minimum and that's going to be where that derivative is 0. Okay, so that's where you have an absolute max or an absolute min. And you'll notice this one has an absolute max but it has no minimum, right? Because it keeps on going as much as you want, you can get as negative as far negative as you want. This one has an absolute minimum, but no maximum. So how do we tell what our function has whether it's an absolute maximum or not? Because we're going to look at some other functions that could have what we call a relative maximum or minimum. Okay, so now what about a function that looks like? Let's see let's make it look kind of like this. Okay, that's a bad sketch. But anyway it looks sort of like that. Okay, so you have sort of a max, It's sort of a min but that's not the absolute minimum because the graph keeps going lower. And this is not the absolute maximum because the graph keeps getting higher. This is an example of what we call a relative maximum and a relative minimum. Okay, so you've seen absolute max absolute min, relative max and relative min. Okay, now this happens to be the approximately the graph of the function f of x is x cubed plus 3x squared minus 9x minus 2. So, how do you tell where these are? Okay, so what's going to happen here, where you get a relative max a relative min, the derivative again, slope of the tangent line, there will be 0. Okay? So we're going to put this together with where the derivative is positive, right means the function is increasing, where the derivative is negative, the function is decreasing and where the derivative is positive. So on this graph, I can tell you that in this area from here over that the derivative is positive, between here and here the derivative is negative and from here over the derivative is again positive. So if I have my function that tells me what functions increasing, then decreasing and then increasing. So that will give me some idea of how to sketch the graph based on the value of the derivative.