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# (a) Suppose that $f$ is differentiable on $\mathbb{R}$ has two roots. Show that $f'$ has at lease one root.(b) Suppose $f$ is twice differentiable on $\mathbb{R}$ and has three roots. Show that $f"$ has at least one real root.(c) Can you generalize parts $(a)$ and $(b)$?

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we want to suppose first impart ate that f is differential on our and has two rooms. We want to show that the first Riveted has at least one. Or in other words, we want to show that there is just some X in the roll numbers where F prime of X is equal to zero. So over on the left here we have kind of a general drawling of what this might look like. So we have two's roots, Um, A and B, where they're both equal to zero at that point. So since it's differential, we know it has to be continuous. And so it between these two were to look something kind of like this. So at some point there's gonna be a horizontal tangent line. Let's just call that it X, and that means is going to have a slope of zero for the tangent, or the derivative is going to be zero at that point. So this is the case of where we apply rolls there. So let's go ahead and apply rolls. They're on to this. So our proof is going to be so sense zero is equal to f obey equal to effort be since a m b r two groups by rules. Here, um, I rolls there. There is some X between those two Other exist and X between A and B such that f prime of X is equal to zero. So we can That's what we're trying to shows We part a little proof box with Smiler Face because we're glad we're done with it. So again, just kind of making No, it doesn't even matter that this was equal to zero right here. The only thing that we really care about is that we have two points that are equal to each other. Ah, and this is gonna kind of be important when we go to part, see and try to generalize what we did in A and B here, so just kind of keep that in mind now for Part P, it says we want to assume that f is going to be twice. Differential has three routes and we want to show the second derivative has a route. So that's really just saying show for some easy in the real numbers that the second derivative of Z is it was zero. So we kind of have the same drawing for F to start. So we know there's three routes, so we'll just call them A B and C. And between A and B, there is something that has a horizontal ask himto Arab horse last 08 horizontal tangent line and likewise between B and C. There should be something with a horizontal tangent line. So both X and Y will be equals zero when we plugged those end to the first rivet. Now, if we were to kind of draw with the graph of first derivative might look like it looked something kind of like there and or at least for the graph, we kind of have right above. And so between X and y, since this is gonna be twice differential, the first derivative has to be continuous. And so again, we could just apply rolls through room and it's going to show us that we have to have this horizontal tanja line. So the second derivative has to be zero at that point. So it's gonna be pretty much we're gonna apply rolls through three times, or we can really think about it as we get to the first derivative case and then just apply party. All right, So proof. So since and we're going to assume that a is less than be is less than the so since, um, zero is equal to f of a eagle to f A B equal to folksy, then by rules there rules a room. There are some ex in between A and B and some Why between b and C such that f prime of X is equal to zero and f prime of why is equal to zero r and now this is the case of part A. So now bye part a sense f Prime has two routes and is differential. And remember, it's differential sense F is twice different trouble, So that means the first derivative has to be defensible. Then there is some Z in between X and y, where f double prime of Z is equal to zero. And so again, just kind of bringing the same point up that we did in the first word. It doesn't matter that these were roots. All we care about is that we had three numbers are three points that are all equal to each other. So now when we come over here to try to generalize it. Um, depending on what they're asking, there's two ways we can kind of do this. So the first way and probably the most obvious for what they're asking is what they said was. So if, uh so if we have some function that has in derivatives, So it's differential. And so in the first case, it was first different trouble. But we had two routes. In the second case, it was twice different trouble and had three routes. So then we're gonna need in plus one roots in plus one roots. So, actually, if instant forgettable on our and in plus one roots, then what was the insta riveted that had that route, then F to the EMTs derivative? Because, remember, we use the parentheses like that to say what derivative it is, has a route, and now the other way that we could actually kind of generalize. This is like what we were saying of. It doesn't matter that it was equal because rolls there, um really only wants that we have the points to be the same. So now we can go ahead and say so if inthe different troubles, that that doesn't change it all differential on or and we have in plus one of the same points and plus one same. Maybe I should say, saying outputs or in other words, we have f X one. This is going to be equal to and let me do this right below. Actually, so f of X one is equal to that of all the way up to f of X and plus one and obviously ex wantto x and plus one are all different. Then we have that f of in has a route. So again, depending on what they mean by generalized A and B, it could be either off these here.

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