Extreme Value

So we'll now t But in all of these theorems, it's always fun to t Why is it laid out the way it is? And that might give us a little bit of more of intuition about it. So the extreme value theorem says if we have some function that is continuous over a closed interval, let'say the closed interval is from A to B. And when we say a closed interval, that means we include the endpoints And B. That's why we have these brackets here instead of parentheses. Then there will be an absolute maxim So that means there exists. There exists an absolute maxim So let's t You're probably saying, well, why do they even have to write a theorem here? And why do we even have to have t And we'll see in a second why the continuity actually matters. So that's my y-axis. And let's draw the interval. So the interval is from A to B. So let'say that t And t Let'say that t So that is F of A. And let'say t So t And let's say the function does something And I'm just drawing somet So I've drawn a continuous function. I really didn't have to pick up my pen as I drew t And so you can see at least the way this continuous function that I've drawn, it's clear that there's an absolute maxim The absolute minim And t And it looks like we hit our absolute maxim And t So another way to say the statement right over here, if F is continuous over the interval, we could say there exists a C and D that are in the interval. So there are members of t For all x in the interval. Just So in this case, they're saying, look, we hit our minim That's that right over here. We hit our maxim And for all the other x's in the interval, we are between those two values. Now, one t And once again, I'm not doing a proof of the extreme value theorem, but just to make you familiar with it and why it'stated the way it is. And just you could draw a bunch of functions here that are continuous over t Here, our maxim At A for a flat function, we could put any point as a maxim But let's dig a little bit deeper as to why F needs to be continuous and why t So first, let's t Well, I could easily construct a function that is not continuous over a closed interval where it is hard to articulate a minim And I encourage you, actually, pause try to construct a non-continuous function over a closed interval where it would be very difficult, or you can't really pick out an absolute minim Well, let'see. Let me draw a graph here. So let'say that t Let'say that's A, that's B. Let's say our function did something right where you would have expected to have a maxim Let'say the function is not defined. And right where you would have expected to have a minim And so right over here, you could say, well, look, the function is clearly approac But that limit can't be the maxim The function never gets to that. So you could say, well, let's get a little closer here. Maybe this n So you could say maybe the maxim 9. But then you could get your x even closer to t 99 or 4. 999. You could keep adding another 9. So there is no maxim Similar over here on the minim There is, you can get closer and closer to it, but there's no minim Let'say that t So you could get to 1. 1 or 1. 01 or 1. 0001. And so you can keep drawing some zeros between the two 1s. But there is no absolute minim Now let's t Why you have to include your end points as candidates for your maxim Well, let's imagine that it was an open interval. Let's imagine an open interval. And sometimes if we want to be particular, we could make t And if we wanted an open interval right over here, that's a, that's b. And let's just pick a very simple function. Let's say a function So right over here, if a were in our interval, it looks like we hit our minim f of a would have been our minim And f of b looks like it would have been our maxim But we're not including and b in the interval. T So you can keep getting closer and closer to b and keep getting Because once again, we're not including the point b. Similarly, you could get closer and closer to and get smaller and smaller values. But a is not included in your set and your consideration. So f of a cannot be your minim So that's one level. It's kind of a very intuitive, almost obvious theorem. But on the other hand, it is nice to know why they had to say continuous and why they had to say a closed interval