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# Extreme value theorem

AP.CALC:
FUN‑1 (EU)
,
FUN‑1.C (LO)
,
FUN‑1.C.1 (EK)
,
FUN‑1.C.2 (EK)
,
FUN‑1.C.3 (EK)

## Video transcript

so we'll now think about the extreme value theorem which we'll see is a bit of common sense but in all of these theorems it's always fun to think about the edge cases why is it why is it laid out the way it is and that might give us a little bit of a more of intuition about it so the extreme value theorem says if we have some function that is continuous over a closed interval let's say the closed intervals from A to B and when we say a closed interval that means we include the endpoints a and B that's what we have these brackets here instead of parentheses then then there will be an absolute maximum value for F and an absolute minimum value for F so then that means there exists this is a symbol for the logical symbol for there exists there exists an absolute absolute absolute maximum value value of f of F over interval over interval and absolute and absolute minimum value of f over the interval so let's let's think about that a little bit and and this probably is pretty intuitive for you you're probably saying well you know why do they even have to just write a theorem here and why do we even have to have this continuity there and we'll see in a second why the continuity actually matters so let's so this is my x-axis that's my y-axis and let's draw the interval so the intervals from A to B so let's say that this is a and this is B right over here let's say that this right over here is f of a so that is f of a and let's say this right over here is f of B so this value right over here is f of B and let's say the let's say the function does something like this let's say the function does something like this over the interval I'm just I'm just drawing something somewhat arbitrary right over here so I've drawn a continuous function I really didn't have to pick up my pen as I as I drew this right over here and so you can see at least the way this continuous function that I've drawn it's clear that there's an absolute maximum and absolute minimum point over this interval the absolute minimum point what seems like we hit it right over here when X is let's say this is X is C and this is f of C right over there and it looks like we hit our absolute maximum point over the interval right over there when X is let's say that this is X is equal to D and this right over here is f of D this right over here is f of D so another way to save the statement right over here if f is continuous over the interval we could say we could say there exists there exists a C and D that are in the interval that are in the interval so there are members of this set that are in the interval such such that and I'm just using the logical notation here such that F of f of C F of C is less than or equal to f of X which is less than or equal to f of D for all for all X in the interval for all X in the interval just like that so in this case they're saying look we hit our minimum value when X is equal to C that's that right over here our maximum value when f is equal to D and for all the other X's in the interval we are between those two we are between those two values now one thing and we could we could draw other continuous functions and 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's stated the way it is and just you could draw a bunch of functions here that are continuous over over this closed interval here our maximum point happens right when we hit B and our minimum point happens at a at a for a flat for a flat function we could put any point as a maximum or the minimum point and we'll see that this would actually be true but let's dig a little bit let's dig a little deeper as to why eff needs to be continuous and why this needs to be a closed why this needs to be a closed interval so first let's think about why does f need to be continuous well I can easily construct a function that is not continuous over a closed interval that where there it is hard to articulate a minimum or a maximum point and I encourage you actually pause this video and try to construct that function on your own try to construct a non continuous function over a closed interval where it would be very difficult or it's you can't really pick out a minute an absolute minimum or an absolute maximum value over that interval well let's see let me draw a graph here so let's say that this right over here is my interval let's say that's a that's B let's say our function did something like this let's say our function did something right where you would have expected to have a maximum value let's say the function is not defined and right where you would have expected to have a minimum value the function the function is not defined and so right over here you could say well look the function is clearly approaching as X approaches this value right over here the function is clearly approaching this limit but that limit can't be the Maxima because the function never gets to that so you could say well let's get a little closer here you know maybe this number right over here is 5 so you could see maybe the maximum value is 4.9 you can get you get your X even closer to this value and make your Y be four point nine nine or four point nine nine nine you could keep adding another nine so there is no maximum value similar over here on the minimum on the minimum you draw a little bit so it looks more like a minimum there is you can get closer and closer to it but there's no minimum let's say that this value right over here is one so you could get to one point one or one point zero one or one point zero zero zero one and so you can keep throwing some zeros between the two ones but though there is no there is no absolute minimum value there now let's think about why why it being a closed interval matters why you have to include your endpoints as kind of candidates for your maximum and values for over the over the interval well let's imagine let's imagine that it was an open interval let's imagine an open interval let's imagine an open interval and sometimes you know if we want to be particular we could make you know this is the closed interval right over here brackets and if we wanted an open interval right over here that's a that's B and let's just pick the a very simple function let's say a function like this let's say a function like this so right over here if we if a were in our interval it looks like we hit our minimum value at a f of a would have been our minimum value and F of B it looks like it would have been our maximum value would have been our maximum value but we're not including a and B in the interval this is an open interval so you can keep getting closer and closer and closer to B and keep getting higher and higher higher values without ever quite getting to B because once again we're not including the point B similarly you could get closer and closer and closer to a and get smaller and smaller values but a is not included in your set and their consideration so f of so f of a cannot be your minimum value so that's on one level it's kind of a very intuitive almost obvious theorem but I did a hand it is nice to know why they had to say continuous and why they had to say a closed interval like this
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