Okay, so we're going to use the intermediate value theorem to show That there is a zero in this function on the interval from 1-2. So the intermediate value theorem says that if we have a value on the interval, say the intervals from A to B. If we have a value of F F. A, that is greater than C, where C is just a constant, so it's equal to some value. And then we have at F f B, we have that F B is less than C. Then as long as this is a continuous, smooth curved function, which this is, then we're guaranteed to have a value at some um value in between A and B. I'll just call it F F. D where D. Is, and I'll even write it down where A is less than D, which is less than B. We're guaranteed to have a value here, that's equal to see. So really, what we need to show is that there is um that F of one is greater than zero or less than zero, since this works either way, and then F of two is the opposite. So if F of one is greater than zero and ff two needs to be less than zero, and ff of one is less than zero than ff two needs to be greater than zero. Then we can guarantee that there is a zero in between that interval. So let's look at F of one, This is equal to four times one of the third minus two times one squared minus seven, which is equal to four minus two minus seven, which is equal to um 2 -7, which is equal to negative five. So now we have that f of one is less than zero, so now we need to to be greater than zero to guarantee that we have a zero in that interval. So it's like that F F two, this is equal to four times two to the third minus two Times two Squared -7. So to to third is eight and if we multiply eight x 4 we get 32 and then two squared is four minus two is eight, and then we have minus 7, 30 to minus eight is 24. And then if we minus seven we get Um 17, so this is equal to 17, so therefore FF two is greater than zero. So if we use the intermediate value theorem, that means that on the interval, so on the interval Um from 1 to 2 we are here guaranteed. Yeah, to have FFC equal zero at some see value. Um in this interval, so as long as C is between one and two, since f of one was less than zero and ff two is greater than zero were um guaranteed to have some value or to have f f c where C is some value equal zero in between that interval.