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# Worked example: using the intermediate value theorem

AP.CALC:
FUN‑1 (EU)
,
FUN‑1.A (LO)
,
FUN‑1.A.1 (EK)

## Video transcript

let f be a continuous function on the closed interval from negative 2 to 1 where f of negative 2 is equal to 3 and f of 1 is equal to 6 which of the following is guaranteed by the intermediate value theorem so before I even look at this what do we know about the intermediate value theorem well it applies here it's a continuous function on this closed interval we know what the value of the function is at negative 2 it's 3 so let me write that F of negative 2 is equal to 3 and F of 1 they tell us right over here is equal to 6 and all the intermediate value theorem tells us and if this is completely unfamiliar to you I encourage you to watch the video on the intermediate value theorem is that if we have a continuous function on some closed interval then the function must take on every value between the values at the endpoints of the interval or another way to say it is 4 for any L what between between 3 & 6 3 & 6 there is at least one C there is at least one 1 C 1 C between or I could say 1 C in the interval from negative 2 to 1 the closed interval such that F of C is equal to L this comes straight out of the intermediate value theorem and just saying it in everyday language is look this is a continuous function actually I'll draw it visually in a few seconds but it makes sense that it is continuous it's a if I were to draw the graph I can't pick up my pencil well then it makes sense that I would have to take on every value between 3 & 6 or there's at least one point in this interval where I take on any given value between 3 & 6 so let's see which of these answers are consistent with that we only pick one so f of C equal tools for so that would be a case where L is equal to four so if so there's at least one C in this interval such that F of C is equal to four we could say that but that's not exactly what they're saying here F of C could be four for at least one C not in this interval remember the C is our X this is our X right over here so the C nu is going to be in this interval I'll tell you I'll take a look at it visually in a second so that we can validate that we're not saying for at least one C between three and six F of C is equal to four we're saying for at least one C in this interval F then F of C is going to be equal to four it's important that four is between three and six because that's the value of our function and the C needs to be in our in our closed interval along the x-axis so I'm going to rule this out they're trying to confuse us all right F of C equals zero for at least one C between negative two and one well here they got the interval along the x-axis right that's where the C would be between but it's not guaranteed by the intermediate value theorem that F of C is going to be equal to zero because zero is not between three and six so I'm going to rule that one out I'm going to rule this one out saying F of C equals zero and let's see we're only left with this one so I hope it works so F of C is equal to 4 well that seems reasonable because 4 is between 3 & 6 for at least one C between negative 2 and 1 well yeah because that's in this interval right over here so I am feeling good about that and we could think about this visually as well the intermediate value theorem when you think about it visually makes a lot of sense so let's let me draw the x-axis first actually and then let me draw my y-axis and I'm going to draw them at different scales because my y-axis well let's see if this is if this is 6 this is 3 that's my y-axis this is 1 this is negative 1 this is negative 2 and so we're continuous on the closed interval from negative 2 to 1 and F of negative 2 is equal to 3 so let me plot that F of negative 2 is equal to 3 so that's right over there and F of 1 is equal to 6 so that's right over there and so let's try to draw a continuous function so a continuous function it includes these points and it's continuous so the intuitive way to think about is I can't pick up my pencil if I'm drawing the graph of the function which contains these two points so I can't pick up my I can't do that that would be picking up my pencil so it is a continuous function so it takes on every value as we can see it definitely does that it takes on every value between 3 & 6 it might take on other values but we know for sure and take on every value between 3 & 6 and so wonder if we think about 4 4 is right over here the way I drew it it's actually looks like it's almost taking on that value right at the y-axis I forgot to label my x axis here but you can see it took on that value in for AC in this case between negative 2 & 1 and I could have drawn that graph multiple different ways I could have drawn it something like I could have done it and actually it takes on there's multiple times it takes on the value 4 here so this could be our C but once again it's between the interval negative 2 & 1 this could be our C once again in the interval between negative 2 & 1 or this could be our C in between the interval of negative 2 & 1 that's just that way I happen to draw it I could have drawn this thing is just a straight line I could have drawn it like this and then it looks like it's taking on for only once and it's doing it right around there this isn't necessarily true that you take on you take on that you become 4 for at least one C between 3 & 6 3 & 6 aren't even on our graph here I would have to go all the way to so 2 3 is 2 no there's no guarantee that our function that our function takes on for 4 1 C between 3 & 6 we don't even know what the function does when X is between 3 and six