 It turns out that not all local extrema are created equal. Let F be a function defined on some domain. Let C be some number inside of that domain. Then F of C is called a absolute maximum. If F of X is less than or equal to F of C for all numbers X in the domain. Likewise, F of C is called an absolute minimum. If F of X is greater than or equal to F of C for again, all X is inside the domain. If you're an absolute maximum or an absolute minimum, then we call this an absolute extremum. Sometimes these are called global extrema. That's also an acceptable term in the situation. An idea is something like the following. If we were to draw a picture, let's just do a simple polynomial function. We get something like say this. Let's find the local extrema. We see that we have a local, here we're going to write this in red here, a local maximum right here at this point. We have a local minimum at this point. We have a local maximum right here. We have a local minimum right here. And we have a local maximum right here. Now, is there an absolute maximum or a global maximum of some kind? That is, is there one number, I should say one value that's greater than every other value in the range? And the truth of it is right here, this is the absolute maximum value on this graph. It doesn't depend on the neighborhood because if we make the neighborhood get bigger and bigger and bigger, the larger the neighborhood gets doesn't change the fact that that number is the biggest, excuse me, that's the biggest value inside the range. That does depend a lot though if we pick different points, right? Like if you take this maximum value shirt on that neighborhood, it's the biggest. And on this neighborhood, it's the biggest. And on this neighborhood, it's the biggest. Eventually though, if we draw the neighborhood bigger, eventually you'll get someone who's actually bigger than this value right here. So maybe this kid was the largest kid, the biggest bully at elementary school, but when he goes off to middle school up, there's someone even bigger. And then when you go off to high school, there's even a bigger bully there too, right? The local, and just why we call them local extrema is depends on the locality. There are certain neighborhoods where it's the biggest, but on other neighborhoods, there might be bigger values in the range. Same thing can also be said with minima here. This is a local minimum because there is a neighborhood for which it is the smallest. This is a local minimum because there is a neighborhood where it's the smallest. This number right here, or this point right here is not an absolute minimum because if we draw a large enough neighborhood, we can find a point that's even smaller. But on the other hand, this point right here is not the absolute minimum either because I can draw a neighborhood that'll have smaller values. Like if I draw a really big neighborhood, you'll notice there's a point over here with a Y coordinate that's smaller than that one right there. And so that's not the absolute minimum either. This graph would actually be an example of a graph with no absolute minimum value. Let's look at one other example here. Let's consider a graph that does something like this. Okay, so this is our function F right here. We can see very quickly the local extrema. There's a local maximum here and here and there's a local minimum right here. In terms of absolute minimum, there is none, right? So there's no absolute minimum, just like we saw before. What about the absolute maximum? It turns out that these two values right here are the exact same Y coordinate. Now this doesn't mean that there are two absolute maximum values. There's only one absolute value, absolute maximum value because the value is the Y coordinate. There's only one Y value. Let's say that both of these are Y equals seven. If there's ever a tie, that doesn't mean there are multiple maximum values, at least not absolute maximum values. There certainly are multiple extreme points. That is, there's more than one point that obtains the absolute maximum value. And so we can talk about there's multiple X coordinates, but there's just a unique Y coordinate. So if a function has an absolute maximum or absolute minimum value, that number, I should say that value is necessarily unique, although there could be multiple numbers in the domain that obtain those absolute extrema if they exist.