 Welcome back. In the previous video, we had talked about when a graph of a function is monotonic and where does it concave? So we're talking about increasing, decreasing for monotonicity. And we also talked about concave upward and concave downward. So related to the notion of concave or related to the notion of increasing and decreasing is the idea of an agreement. So imagine we have a function f and we look at some number inside the domain of the function. So this is in particular going to be some x coordinate right here, x equals c. We say that the associated y coordinate f of c is a local maximum if for f there exists some open interval a to b that contains c and satisfies the inequality f of x is less than or equal to f of c for all x's in that interval. So a quick picture of something like this would be like the following. Our function maybe does something like this in which case we have this point associated to that point is x equals c. Well then it's like hey there exists some other points x equals a on the left. Another point over here x equals b on the right. In which case you'll then notice that inside of inside of this picture right here this is the this is the biggest value in that picture that small little thumbnail of the graph. We're not looking at the entire graph which is why it's referred to as a local or sometimes referred to as a relative maximum. It might not be the biggest point on the entire graph but the y coordinate is biggest for some neighborhood that surrounds x equals c. That's what we're looking for with this idea of a local maximum which will then satisfy the inequalities we see right here. The f of x will be less than or equal to f of c so no one in the interval will be bigger than f of c. And so analogously we can define the idea of a local minimum or sometimes people call it a relative minimum. Same basic idea but turned upside down. A relative minimum of f is going to be some y coordinate f of c such that in some neighborhood around c everything is never smaller than it right. So if we look at our function again so we have our point right here c comma f of c. And so what we're saying then is that there's some interval some neighborhood we oftentimes call it. We could draw a little circle around the point and in that circle there's some circle for which that point is the smallest thing present. No one can get smaller than that. And so this gives us a local maximum. We say that a point is a local extremum if it's either a maximum or a minimum. So we can see we can think of extremum as this neutral term not specifying a maximal or minimal whatsoever. Now a bit of grammar I do want to mention here is that words like maximum, minimum and extremum do come from Latin. And so the proper way of making this a plural word would actually be like extrema, minima, maxima not maximums. Mums are flowers. Maximum is the proper one. I mean obviously no one's going to get like taken away by the grammar police if you say that incorrectly. But I just want to point that out to you here. Now closely related to the idea of an extremum is the idea of a point of inflection. So the reason why these things are related here is I want to point out that an extremum is a point on a graph where the function changes its monotonic behavior. The maximum is a place where the graph switches from increasing to decreasing. And it's going to happen here at this maximum value. Likewise a minimum is going to be a place where the function switched from decreasing to increasing and you see something like this at the minimum value. So extrema are points on the graph, the local extrema are points on the graph where the function switches its monotonicity. We define similarly the idea of a point of inflection or sometimes they're called inflection points. An inflection point is a place on a graph where the function changes its concavity. So you might see something like the following. Your function was concave downward right here then it switched to be concave upward and thus you get this point of inflection, private eye right there. Another example you might see something like the following. Same basic idea you switch from concave upward to concave downward and thus you get this point of inflection right there. And so we want to be able to visually identify the local extrema points of inflection. So let's return to our friend from before. We saw this graph in a previous video. So let's first look for the extrema on this graph. This is something we can identify just by inspecting the graph. So first of all, there are some sort of obvious candidates here. So the first one I want to point out here is going to be the point negative four, negative two. This right here represents a local minimum on the graph. Because you can see that in this neighborhood, no one is smaller than it in that neighborhood. So that's what we call a local minimum. Some other points that I want to throw out there is that if we look at the point negative six zero, this right here will also be a local extrema. This is an example of a local maximum. This is a local maximum. The reason we call this I mean a local maximum because in that neighborhood, no one gets bigger than negative six zero with respect to the Y coordinate. And that's kind of it's kind of a little cheating because the fact that the point is on the boundary of the domain of this function, it makes it a lot easier for it to be the biggest point. If you're a boundary point on the left and your functions decreasing, you automatically be a local max and be fine. With negative four, negative two, we see that it's a minimum because it was decreasing then it switched to increasing. That change in monotonicity gives us a local extrema. And so in that vein, we also can see that six one would likewise be a local minimum. Because in that neighborhood, that blue little circle I drew on the screen, no one gets smaller than Y equals one in that neighborhood. So that gives us a local minimum. And you're going to see that if your function's decreasing and then suddenly stops, that endpoint is going to be a local minimum. But it turns out there's a lot more extrema on this graph right here. Some that might be some ones we could still guess right here is if you take the point three comma four, this is going to be a local maximum. This is going to be a local maximum because in the neighborhood that I've drawn here, no Y coordinate gets bigger than Y equals four. So that's a local max. And so by similar reasoning, you also get this local max here at zero comma four, because again at that point, no one gets bigger than it. And so if we keep track of some of these things here, our local minima remember we had this point at X equals negative four, and at X equals six. So when you're asked about extrema, there's there's the coordinates you have to pay attention to, right? When you're asked about like a minimum, are they asking for the X coordinate or are they asking for the Y coordinate? We have to be careful about that. Because the minimum itself is the Y coordinate. There's a local minimum right here. But we often will refer to the point by its X coordinate. Because as it's a function, there could be multiple places where the graph is negative two, Y equals negative two, there could be multiple places where the graph is Y equals four. And referring to a extrema, you typically are going to refer to its X coordinates like the minima are at X equals negative four, and X equals six. What about the maxima? The maxima are going to be at X equals negative six. It's going to be at zero. It's going to be at three we mentioned, but it turns out there's more maxima on this graph. You might be a little perplexed at the moment like really, what else is still going on here? It turns out that every point between zero and three is actually considered a local maxima. Because it's considered a local maxima. Excuse me. The grammar police are going to come after me very soon. Break down my door. The every point between zero and three is a local maxima. Like if you take X equals two for a moment, if you draw the neighborhood around X equals two, is it not true that every point between X, every point around X equals two, there's no Y coordinate gets bigger than it, right? Remember, we come back to the definition of a local maximum. This was a less than or equal to symbol. To be a maxima just means that no one gets bigger than you. It doesn't mean that you're bigger than everyone else. That's a slight difference in terminology there we have to pay attention to. You'll notice that every point at X equals two, that no point near X equals two is actually bigger than X equals two. It turns out we actually do get a local maximum at X equals two. We should actually record that on the graph as well. You would get X equals two. In fact, you get maxima for any point between three and zero. We can actually improve upon that. There's a maxima negative six for X and then there's a maxima between all points from zero three, where zero and three are included in those maxima as well. That actually opens up a Pandora's box for us right here, doesn't it? X equals two gives us the point of a local maximum, but it also gives us the point of a local minimum because when you look at that red circle, there's no point that's smaller than it either. Huh, that's kind of weird. So what? Every point between zero and three is also a local minimum? The answer is yes. It's also true that when X is between three and zero, you'll get a local minimum. Because again, at that point, nothing's bigger, nothing smaller than it in that neighborhood. And this is something you're going to happen. You're going to see on a constant interval. If ever a function is constant on some interval, then all points in that constant interval will be both local maxima and local minima. And that's the only way that a point can be simultaneously a maximum and a minimum. And so we're not going to see constants happening on a function that often. And so be aware that when it comes to extrema, this is somewhat of an abnormal situation. You see things like right here where it's strictly increasing or strictly decreasing. But if it's constant, weird things are going to happen in terms of the modernity. Points are of simultaneously minimum minima and maxima. Now I should mention that the point at zero and the point at three, these points are not actually minima. They are genuine maxima. Because while there's no one bigger than it in the interval, which makes it a maximum, there are points that are strictly smaller than it. And therefore, they aren't minima, but they are maxima. So that's kind of a weird thing to say there, but that's what's going on. If we ask the question about points of inflection, where can inflection points be found? Inflection points are going to be those points where the concavity changed. So notice right here at negative two, it switched from concave upward to concave downward. So we get a point of inflection right there. You get a point of inflection right here also at zero, because at zero it switched from concave downward to flat to straight. And then you likewise get another point of inflection right here at three, because it switched from being a straight curve to a concave upward curve. And so you see points of inflection there. So it's possible that an extremum could also be a point of inflection. And that's particularly going to happen when all these weird things are going on with the straight line there. Now I wouldn't call any of the points between zero and three points of inflection, because the entire time it stayed straight the entire time. And so by the definition we get the following terminology.