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