 Welcome back to our lecture series Math 12-10 Calculus 1 for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. This is the first video for Lecture 32, which is going to be very theoretical in its approach. That is, we're going to be talking about some theorems and proofs that are going to be important for our study of the derivative, and particularly we want to build up to the so-called mean value theorem, which is one of the most important results from differential calculus. Before we state the mean value theorem, we actually are going to prove a state and prove a stepping stone to get to the mean value theorem, what's known as Roll's theorem. Roll's theorem is named after the French mathematician Michel Roll, who proved this result in 1691. Basically, it kind of makes sense, but it does require an argument to establish why this would hold for every single differentiable function. Roll's theorem, if f is some function that satisfies the following three conditions. So the first condition is that we want f to be continuous on the closed interval a to b. Second, we need that f is differentiable on the open interval a to b. And thirdly, we want that f of a equals f of b. Let's try to unwrap this a little bit. The last assumption, f of a equals f of b, this just means, kind of like we have depicted on this picture on the right here, f of a equals f of b just means that at the start of the end point and at the end of the, excuse me, at the start of the interval and at the end of the interval, we have the same y-coordinate. So this function starts and stops at the same y-value, although the x-coordinates are certainly different from each other, a and b do not equal each other. All right, that's what the third condition says. It starts and stops at the same y-value. Continuous, right? A function is continuous. Essentially, if you can draw it without picking up your pencil or pen, more specifically, a function is continuous of every place in its domain. The function evaluation agrees with the limit. Differentiable. Differentiable means that the derivative exists everywhere, that the tangent line exists everywhere in these domains. And you will notice a slight difference in the intervals. We expect that f is continuous on the closed interval a to b, but only has to be different, but it's differentiable on the open interval a to b, which is a slightly weaker statement. Now, one has to be careful, right? Because differentiability actually implies continuity. Differentiable is a stronger condition. Every differentiable function is, so why do I need to state both things? Well, that's because that we'll be differentiable between a and b, but at the end points, a and b, we don't make any assumptions about differentiability, but we do still require that it's continuous on the end points. So everywhere between a and b, it needs to be differentiable, but at the end points, it needs to be at least continuous. If it's differentiable at the end points, that's even better, but that's to satisfy Rohl's theorem here. So in summary, if we have a function which is continuous from a to b, inclusive, because again, the brackets mean that a and b are included in the interval there. If f is continuous from a to b inclusive, if f is differentiable between a and b and if f of a and f of b are the same thing, then there exists some number c, where c is going to live between a and b, although we do know that c will not be a, nor will it be. So c will be in the open interval, it won't be in the closed interval necessarily. That is, it won't be the end points. If under these assumptions one, two, and three, then we know there's some point c such that f prime of c is equal to zero. That is, we're guaranteed at some point, the derivative, now the derivative of zero, that would suggest the function has a horizontal tangent line at that point. And so when you look at this diagram here, that intuitively seems to make sense from a geometric point of view. It kind of, the idea is the following. If you have some point over here and some point over here, if I'm drawing a, if I'm drawing some type of continuous differentiable function, basically what goes up must come down. If there's no breaks or sharp corners in the graph, then yeah, there's gonna have to be some place where there's a horizontal tangent line. You could interpret this, not just as functions, but also in physics, right? You could think of this as f is the motion function, the position function of some particle that's moving. If ever there's a moment where the particle has the exact same position at different points of time, mind you, but if there's some point t equals a and t equals b, it has the exact same position function that is f of a equals f of b here, where a and b are time stamps, time t here. Then Rolls-Therm says that there is some instance, some instance between the two at time equals c. Again, this is between a and b such that f prime at c is equal to zero, which if this is our position function, the derivative of position is equal to velocity. So we see the velocity of c is equal to zero. So the only way that a object can move from a position back to itself, assuming it left that position, it would have to at some point have zero velocity. There's some point where it comes to a pause. So if you throw like a ball in the air, it goes up, it comes down. At some moment in that trajectory, the ball has no motion. It stops. It has no velocity. All right. So again, intuitively Rolls-Therm seems to make sense, but let's this logically speaking, that is, let's consider the proof of Rolls-Therm. And it turns out it's really not so bad based upon essentially the discussion of critical that we saw in the previous lecture. So the first situation is what if our function's a constant? If it's a constant function, then it's flat. This is our function f right here. Well, the derivative of a constant function is exactly zero. And so Rolls-Therm says there's at least one point where the derivative would be zero. If your function's constant, then every point between a and b will have zero derivative. So you have lots of options for c. That's an important case to mention here because when we talk about Rolls-Therm or the subsequent mean value theorem, these theorems guarantee the existence of certain points with conditions, but it doesn't say that point's unique. There could be, right? It could be that our function goes up and goes down a couple of times, right? There could be multiple horizontal tangent lines between these points that have the same y value. That's okay. For a constant function, every point would have a derivative equal to zero. All right. The second case is a little bit more interesting. Let's suppose, much like I'll try to keep it on the screen for a little bit longer, let's suppose like our picture suggests, let's suppose that there's some y value between a and b so that it's bigger than f of a. Okay. So let's draw that picture for a moment. Let's say that we have here our f of a. We have our f of b. And again, these have the same y value. So this is a horizontal line right here. Let's suppose there's some point, some point f of x where x is between a and b, that f of x is bigger than f of a and necessarily larger than f of b, which is perfectly fine. So there's something that gets above this horizontal line here. Well, by the extreme value theorem that we saw previously, this is why continuity is important. Coming back to our assumptions, we're assuming that f is continuous on the closed interval a to b. This is exactly the setting where the extreme value theorem applies that we talked about in the previous lecture. The extreme value theorem says if you have a continuous function on a closed interval, at some point on the interval a to b, you will obtain an absolute maximum value. Likewise, they'll be about the maximum right now. So the extreme value theorem guarantees there exists some c, which obtains the absolute maximum of that interval. So f of c is greater than equal to all of the f of x's in the situation. Well, since there's some f of x that's larger than f of a, then it turns out that f of a cannot be the absolute maximum because there's something bigger than it. And since f of b is equal to f of a, then f of b cannot be equal. And so we see that this c value cannot be equal to a, cannot be equal to b. It's got to be something strictly between f of a and f of b. And so that's what we're going to take here, this absolute maximum. We'll say right here, this is our f of c value. It's our absolute maximum. And it's distinct from the a and from the b because they can't be equal to that. Well, we also want to apply Fermat's theorem. Fermat's theorem says that if you have an extremum such as an absolute extremum, then one of two things must happen. The extremum must be at an end point, but nope, it's not at f of a, it's not at f of b. Or if it's not an extremum, then it has to be a critical number, which a critical number means that the derivative at c is either undefined, d and e, or it's equal to zero. But the derivative is defined because the function is differentiable between all points a and b. Therefore, f prime of c exists. And so by Fermat's theorem, we see that it's going to have to equal zero. So there has to be a horizontal tangent line at this value x equals c. Now similarly, if f of x is, if there's some f of x that's less than f of a, a similar argument can be used here. The latter phrase is mutatis mutandis, which basically just says change the appropriate parts. So the main difference is when we invoke, we look for a absolute minimum as opposed to an absolute maximum. And then you apply Fermat's theorem, which will then show that the absolute minimum would have to be equal. So Roll's theorem is then essentially just a consequence of the extreme value theorem and Fermat's theorem. In the next video, we'll modify Roll's theorem to give us the mean value theorem.