All right, welcome back. Let's talk about the intermediate value theorem. As with any theorem that has a name, of course, that means it's probably important, so we should know it, here it is, statement. This is again called the intermediate value theorem. Sometimes we abbreviate this as just IVT, IVT Intermediate Value Theorem, and it says the following. Suppose that f is a continuous function, and this is super-important, this word "continuous", on a closed interval a to b with f of a less than or equal to some number n less than or equal to f of b. Let's pick it such that where f of a is not equal to f of b, here comes the result. All of those conditions are true then, there exists some real number c such that f of c is equal to n. Stare at this for a second, one more time. Suppose f is continuous on a closed interval a to b with f of a less than or equal to n, less than or equal to f of b with f of a not equal to f of b, then there is a c such that f of c is equal to n. What does that all mean? Whenever you get something wordy in math, draw a picture, try to put a geometric intuition between it. Okay, so here we go. First off, I have a to b, and I have some function that's continuous, very important, it's all about continuity on this interval. Draw some nice function with a continuous whatever does its thing. No gaps, no asymptotes, no nothing. The output, of course, here's f of a and here's f of b. They live on the y axis. Now, here's what it says. It says, pick any number n, anyone you want somewhere on the y-axis. It says, if you draw the function, if you draw like a line across from n, you have to have, there must exist a point where this line from n touches the graph. Where it touches the graph, that x value is c, such that f of c is equal to n. I want you to try this if you want. Try to draw the graph I drew here. Just some basic one, is pretty simple. You may say, well, can I draw a more complicated one that goes up and down. Super-high, super low does crazy stuff. You can, but I guarantee you will have to cross this blind I have here in green, f of c equals n. You will have to cross this line at some point. This statement is really about how functions grow, if you think about just as some analogies here, if I told you that a person was 15 years old and that a person was 20 years old. There has to be some time when the person was 18 years old. You can't skip because time is continuous, age is continuous. It tells you about how things grow. However, if I said, "well I have an elevator and it stopped on the first floor and it stopped on the fifth floor." It didn't have to stop on the third floor. Now it had to pass a third floor sure, for distance and stuff, but it didn't have to open the doors. Something like that. If I'm shoveling gravel into a bag and the bag was two pounds and now it's three pounds. It didn't have to, at any point in time, be 2.5 pounds. It doesn't grow continuously when you do shovels into a bag or something like that. But if you have time, we have age, we have distance. If I drove from New York to Florida, and I drove straight [inaudible]. At some point I had to drive through Baltimore, Maryland or something like that. You can't just jump and skip these things. It's about how things grow, it's about how they move and the idea of continuity is extremely important here. If you take away continuity, if you're allowed to have something that is discontinuous, you could jump over the line f of c equals n, and have a discontinuous function. There's really three things that you sort of need here, one unique continuity, you absolutely need continuity. Let me just give an example why you need this. If this function is not continuous and I have all basically the same setup. I have a to b and just pick an n here. Here's f of a, here's f of b, something similar. F of a, f of b, and I pick an n right there in the middle and I draw this dashed line going across. I can, if I have a discontinuous function, heck, why not think of the greatest integer function. I could jump over it. I can certainly just jump over this thing. Now I can pick up my pencil and never touched the graph. You absolutely need continuity for this to work. The other thing that's a little subtle, but you absolutely needed to work as well is the close interval piece. You need the interval to be closed. The reason for that, so let's just say if the interval a to b is not close, so you'd have an open interval, the parentheses are something like that, you can have asymptotes. You can have asymptotes, here's a, here's b and you can have asymptotes here where the function doesn't let's say you have f of a defined, f of b defined, but the function itself, maybe it goes like this or something. Here's my line n going across. You can just have function arc up and down. If you notice I missed the line, so there's f of a and I defined f of b, for this some piece wise function or some like that. F of b. I defined f of a I defined f of b, but I never quite touch my asymptote. Here's discontinuities, there's asymptote. You can't have open intervals as well. You need this thing, you need this function to be defined on a closed interval, so that you guarantee continuity at the endpoints. Then f of a, not equal to f of b, that's sort of the trivial case. It doesn't come up, it's not as important to talk about, but you just want to have some value of n that's in the middle. One thing to note, this isn't a bad thing or a good thing. It's just a thing. There's nothing stopping me if I had drawn just as another example, like some other function that crossed a couple of times. Just as a note on this example, you may have maybe more than one value of c. Like this first part where the graph intersects would be a, c and a second part would be, these are all perfectly good values, where f of c is equal to n, where there's multiple inputs for the output I want. There may be more than one value, so you can have infinitely many in fact, if you had something that oscillated a lot, but you may have more than one value of c. That's not a good thing, it's not a bad thing. The theorem just says there exists at least one. You could think of it this way. At least one doesn't guarantee the number at all. Just says there's at least one. I guess note number 2, is that while this theorem, guarantees the existence of an input to give you your given output. I can only tell you that it exists. I cannot tell you where it is. It does not tell you what c is equal to. It doesn't find c. This is very like theoretical. It's like there exists. Well, where is it? I don't know if there exist. Like, there exist a hippopotamus. Where is it? I don't know. They're out there. You got to go find it. That means, there are ways to go calculate this thing, but we'll talk about those later. For this theorem only tells you that there exists something. Sometimes when you're doing math, that's all you need. That's all you're asking for. Just as an example, so let's actually use this thing now an example. The questions that you'll see about this, theory, they're interesting, they're theoretical. Here's a usually the classic example with this, take a cubic polynomial. Remember cubic means degree three. You have ax cubed plus bx squared plus cx plus d. Just assume that a is positive, just so we have our classic example, but of course all of this works. Maybe it does something like that. It doesn't have to go through the origin of course, maybe you have something like this. Whatever, pick your favorite degree three polynomial, depending on what a, b, c and, d are, it moves the graph around. Usually go low to high if a is positive. If you notice, whenever you draw this thing, if you draw a bunch of these graphs, you'll always cross the x axis and at least one point, this one has three. Where you have your x-intercepts are the roots of the polynomial. The claim, is that any cubic polynomial has at least one real root. Any cubic polynomial has one real root. There exists at least one real root. It doesn't tell you, where it is, doesn't tell you, what it is, just says there exists one. This feels like intermediate value theorem. Let's go through that. Couple of things we need. One is a polynomial, f is a polynomial of degree three cubed polynomial, so f is continuous. On all reals, now, all reals, you can think of this as minus infinity to infinity. For technical reasons, this is actually both open and closed. Sometimes they call it copen, which is funny, so it's closed. We meet our two conditions that have a continuous function on the number line. Let's pick a number, so we know that the limit. Let's do this first. The end behavior, the limit is x goes to infinity of f of x is infinity. Again, this is for a, greater than zero. It goes up forever, and the limit as x goes to negative infinity of f of x. What happens when x gets really small? The graph goes down. If, a is negative, just reverse the things, but it all works. The point is I have some number down here. It's got to go negative, it's got to go positive. Now treat these as your outputs here, f of a and f of b, the endpoints. What does that tell you? Let's pick before we go on. We'll pick N to be zero. Pick N to be zero in our thing, because we're after a root. Remember, N is your output. If I want a root, if I want to find a root, roots in a row plug in to give me zero as an output. Pick N to be zero. Do you agree that zero is between negative infinity and infinity? Hopefully, you do, so it's not a problem. I have a continuous function on a close interval and I have an N between the two bounds of the interval, some negative number, some positive number. Again, try to draw. You can't. If you're keeping your pen on the paper, you can't go from load high without crossing the x-axis at least once. If I pick N to be zero, it's a little weird because I'm not really calculating anything, I'm more trying to convince you. I'm telling you a story. This is getting to math proofs a little bit. But pick N to be zero. Remember, you can only use IVT if have continuous, close interval, and a value of N between the outputs, check, check, check. By the IVT, there exists some number c on R, so some real number, such that f of c is N. What's N? F is zero. Basically, you have f of c as zero, and therefore c is a root. It's a real root. Every cubic polynomial has at least one real root. Now, there could be more. Remember, intermediate value theorem only says you can have at least one. There could be three. There could be one. If you know a little bit about complex numbers, the roots, they come in pairs, so you can't actually have two but that's sort of an aside. If you have a cubed polynomial, you'll always be able to find a real root. There are methods to find it, but this is just something. If you make this argument for not just cubics but anything with odd degree, so degree five, degree seven, degree nine, you get the same results. There's always at least one real root and this matches. If you know about complex numbers, they'll be the other roots. You're always going to have three total. If you have one real root, you get two complex numbers. If you have three real roots, you had no complex numbers. But you can't have two because complex roots come in conjugate pairs. This is one of the first applications you're going to see. Again, it doesn't tell you where the roots are, I have no idea. I can point to them on a graph but I'll need some calculator or calculation to find the roots itself. Let me give you one more example, and here's the question for you. Is there a number exactly one more than its cube? I want you to pause the video and think about this for a while and just try to see if you can figure it out. This is existence theorem. Pause the video, think about it. Okay, ready? Is there a number exactly one more than its cube? There's lots of numbers out there, so checking each one is probably a bad idea. Let's take this wordy question and put it into some math. Is there a number, so does there exist some x, one more than its cube? I need a number x such that it is one more than its cube. So x plus 1 equals x cubed. Is there a solution? Basically, the question if we rephrase is, does this equation have a solution and a real solution as well? Well, a little rearranging here. It's x_3 minus x minus 1 would equal 0. This question says, is there an x that makes this true? Wait a minute, x cube, this is a cubic polynomial. Therefore, by the intermediate value theorem, by the thing we just did, it must have a real root. Must have some x inside of R that makes this true. There's some x that I plug in and I will always get zero. Such that x cubed minus x. But now maybe there's more than one, I don't know. We'd have to graph this thing. But the big answer, the takeaway of this, is like this question that doesn't seem to relate to what we just did is absolutely related. It's just rephrased. So the answer here is yes. This is a big yes. Now, of course, a bit on the follow-up questions like, well, what's the x? X equals, and you don't get that from IVT. You can go graph it, use a calculator and chase decimals around and solve this thing, but you don't get it from IVT. This is a tough one. If you ask people this, they stumble a little bit. Again, most people are so used to calculating, finding x equals seven or some number that these questions throw them off because they're just a different way to approach. You're telling me a story, you're convincing me something is true without doing necessarily a calculation. Go over these two examples, the other one's in the book, and practice these as you go through. Make sure all the conditions are met and whenever you use a theorem, say by the intermediate value theorem or if you want abbreviate, IVT, that is fine too. Good job. See you next time.
