 Welcome back to our lecture series Math 1210, Calculus I for students at Southern Utah University. As usual, I'll be your professor today, Dr. Anjana Misildine. In this lecture number 31, we're going to start our chapter 4 in this lecture series, which will focus on applications of the derivative. We've seen some applications already, particularly with related rates and rates of change in science and such. Those appropriately could have been put in chapter 4, but we did those earlier in postpone chapter 4 for a different set of topics, mostly to do with the exams schedule that might accompany a lecture series like this one right here. It's somewhat of an arbitrary choice. But what we're going to talk about in chapter 4 is many applications and many of which are going to be derived upon no pun intended there. Most of these ideas are going to be derived from this following lecture about maximum and minimum values on a graph. This is a topic we've introduced previously from a pre-calculus point of view, but it turns out studying maximum and minimum values is most appropriately done in a calculus perspective. We're going to see that in subsequent videos in lecture 31 and in some of the other, the subsequent lectures in this lecture series. Let's first remind ourselves, what are the definitions of these things that we may or may not have seen before. Let's see be a number inside the domain of some function f. C of course can be an x-coordinate for some number inside this domain of f. Then f of c, it's associated y-coordinate is called a relative maximum. Sometimes people use the word local maximum. It doesn't really matter. Local maximum and relative maximum, same thing here. A local maximum for f, f of c exists if there's some open interval a to b that contains c. We need unravel this to understand that c is somewhere between a and b. C is inside that domain. We say that f of c is a maximum if f of x is less than or equal to f of c for all x's inside of that interval. Let's try to draw a picture to understand what's exactly going on right here. Let this be our x-axis. We'll even label it so it's very clear that we have the x-axis. We have then our function. Let this be our function f right here. We claim that this number right here, f of c coincides with a relative maximum of the graph. Well, the way we are going to interpret that is, is let's take an interval that surrounds this number c. A is a little bit less than c, b is a little bit larger than c, and then let's look at what happens on the graph. We have these points right here. In terms of the y-coordinate, this is f of a. In terms of the y-coordinate right here, this is going to be f of b. I'm going to draw this little bubble. Going from f of a to f of b, and this is often referred to as a neighborhood on the graph. It doesn't include the whole things. If we couldn't see anything outside of our bubble, we can see nothing outside of the neighborhood. We just focus on that sector right there. What do we see about the graph? Well, there's something special about this point f of c right here. Because the point is, whether you're a little bit to the left or a little bit to the right of f of c, you're always smaller than f of c. So like we were saying earlier, f of x is less than or equal to f of c for every number inside that neighborhood. This open interval a to b, we're doing Mr. Rogers' neighborhood right here, f of c is the largest number inside of that neighborhood. That's what we define to be a local maximum. The idea of a local minimum is defined similarly, a relative minimum. The idea is, as the same, if we take some neighborhood of the function, and if you have some point that's smaller than every other point in the neighborhood, then we would call that a local minimum. So this right here is a local minimum. This is an example of a local maximum inside of the graph. With this local minimum, you see that f of c is less than or equal to f of x for all numbers x inside of that neighborhood. So we get this local minimum. Now we're gonna interested in local minimum and maximum. Let me point out a little bit of the grammar. Here, maximum actually is a Latin word. And so you don't say maximums, mums or flowers. The plural of a maximum would actually be maxima. Plural for minimum would also be minima. I mean, if you do say maximums, I know it's gonna scold you for horrible grammar, but it is incorrect grammar, as you mentioned that. So the plural of maximum is maxima. The plural of minimum is minima. If we want to choose the gender neutral term, we could talk about extremum. An extremum is a local maximum or a local minimum. And likewise, using the Latin here, the plural of extremum would be extrema. So you're often gonna hear me talk about extrema. And that's really what chapter four is all about. It's finding and using the extrema of a graph to help us understand more about the function and its applications and things like that. Some other things we should point out about extrema is that if you have some type of like half interval, like for example, if you were to take this point on the very end of the domain of our picture, and let's say our neighborhood is looking something like this. Well, what if the graph just doesn't go past a certain point? Yeah, this endpoint. This endpoint would still in this case be considered a local minimum. Because in that neighborhood, no one got smaller than the point on the right, so it makes it a local minimum. On the other hand, this point on the right, this would be an example of a local maximum because endpoints can be extrema. They can be the maximum or the minimum. If you look at this neighborhood surrounding the endpoint, there is a neighborhood, there is some bubble we can draw around the maximum there to show that no one's bigger than that. And do pay attention to these extrema here. Notice how this is a greater than or equal to or less than or equal to. To be a maximum or a minimum, you don't have to be bigger than all the points in the domain or in that neighborhood. You just can't be smaller than anyone else. All right? So for example, if we were to consider this portion, like if we take this point right here and we consider a neighborhood, when you look at this neighborhood, right? There's points bigger than it and there's points smaller than that. That doesn't seem to suggest it's an extrema at all. But maybe you just shrink the neighborhood down, right? But the point is, if you take some point away from these extrema here, no matter how small you make the neighborhood, there's always gonna be someone bigger and someone smaller, so it's neither a maximum or minimum. This gives us the idea of what we mean by these extreme values. Now, a little bit of vocabulary that I do wanna throw out for us here is the difference between the quantities, quantities that we refer to as a number and quantities that we refer to as a value. Because this can get a little bit confusing when we're talking about these extreme values, these extreme points. Am I asking about the x-coordinate or the y-coordinate or whatever? Whenever I refer to a quantity as a value, we will be referring to a y-coordinate. That's not a rule that necessarily everyone who teaches calculus follows, but in this calculus series and in most calculus series I would suggest that when people talk about values, we're talking about y-coordinates. On the other hand, when we use the word number, a number will always be referring to as an x-coordinate. So you might hear me say something like, let's find the number which obtained its local maximum value, okay? So the maximum values are describing the y-coordinates, since this f of c. On the other hand, x-coordinates, we'll call numbers, that'll be like c or a or b in this context. We can also put them together. Sometimes we care about that. We might talk about a maximum point. A point would mean that you have the x and y-coordinates. So you put them together. So be very cautious about when you do these problems on your own or your homework system. If you're asked to find the maximum value, you want the y-coordinate. If you're looking for a maximum point, you want the x and y-coordinate. If you're asking for the number which obtains the maximum or the minimum, then you're looking for the x-coordinate. So take everyone's favorite function, y equals x squared, the standard parabola, which you can see illustrated here on the screen. Does this graph have any local minima or maxima? And we can see in fact that there is a minimum value right here at the bottom of the graph, the so-called vertex of the parabola. The reason why we care about the vertex of a parabola is because it's somewhat distinguishable from the rest of the points on the graph. This is in fact a local minimum value. Of course, if you were to reflect your parabola, so it concaves downward, then you would have a maximum value. And in a precalculus or college-adjust setting, oftentimes efforts are made to search for the minimum or maximum values of the graph. And that coincides with finding the vertex of this parabola. And so, again, in college-algebra settings, we often use quadratic functions as an introduction to the problems about optimization because we can find this minimum value purely from an algebraic point of view, which we'll see later on that that can be difficult to do for general functions, but the derivative is a very effective tool to help us solve these in general. Let's switch this over now to the absolute value function, y equals the absolute value of x, whose graph makes this v-shape here on the screen. It does sort of have that upward direction that the parabola has. And our eyes do gravitate towards this local minimum that we see at the bottom of this absolute value function. When studying absolute value, this is likewise called the vertex. Why do we care so much about this point on the bottom of the graph? This point, zero, zero, assuming there's no transformations made to the graph whatsoever. Why do we care about the vertex of the absolute value function? Just like the parabola, it comes from the fact that this point is important. It's the only extrema of the graph. There's only one extrema, and it's this vertex of the absolute value. It's significant. But one thing I should point out that's different about this minimum compared to y equals x squared is that with this graph, it comes to a point. It's got this sharp little corner on the bottom of it as opposed to the parabola, which is more smooth. We've seen previously that if a function has a sharp corner, the derivative is undefined at that point. And so even though the function's not differentiable at the vertex for this absolute value function, we still do have this local minimum. And so if we're gonna be searching for the extrema of a graph, it turns out that there's something that needs to be said about points like this. Let's switch to a different function. What if we do y equals sine of x? Let's turn to metric function. This function has some extrema, right? It has a local maximum right here, has a local minimum right here, but here's another local minimum. And in fact, if we zoom out on the graph of y equals sine of x, you can in fact see that there are several extreme points on the graph. We have a maximum, a maximum, a maximum, a maximum, a maximum, a maximum. Here's a minimum and a minimum and a minimum. It just goes on and on and on and on. We have this maximum here and here and here and here. We also have minimum here and here and here and here. This sinusoidal wave, in fact, has infinitely many extrema. It has infinitely many maximum values and has infinitely many minimum values. For sine with no transformations applied to whatsoever, the maximum always happen at y equals one, and the minimum, the local minimum values will always happen at y equals negative one. And we see this happens over and over and over again because of the periodic nature of sine of x. As one last example, let's consider the function y equals x cubed. When you look at this graph, you'll notice there's not any obvious maximum or minimum values. In fact, as we zoom out of the function, it just goes on the right-hand side up to infinity. On the left-hand side, it goes off towards negative infinity. This function has no absolute maximum or minimum. Of course, when we look at the graph, it feels like something important is happening at the origin x equals zero, y equals zero. It's not an extremum, though, because there are values bigger than it and there are values less than it. It's not gonna be a local maximum or local minimum. So what's significant about that? It turns out we're gonna have to return to this on some future date.