 Hi, I'm Zor. Welcome to Unisor Education. Today we'll talk about analysis of certain smooth functions using their derivatives. Well, this lecture is part of the advanced course of mathematics for teenagers and high school students. It's presented on Unisor.com and that's where I actually suggest you to, which this lecture from, because every lecture has detailed notes which basically serve as a textbook, plus registered students have ability to go through some enrollment, exams, and it's free, by the way. Okay, so maximums and minimums of functions. Well, first of all, as we discussed before, if a smooth function, now smooth, let me just repeat a little bit, when I'm talking about certain derivatives, I assume that the function does have a derivative. If I'm talking about derivative of a higher order, like second derivative, for instance, again, I assume that the first derivative is differentiable to produce the second derivative. So, when I'm talking about smooth function, it basically a replacement for saying all the time, well, considering this particular derivative exists, et cetera. So, smooth means whatever I'm saying does take place and available for this particular function. So, a smooth function, if it has a local minimum or local maximum, so this is a local minimum and this is the local maximum. Now, as we know from the Fermat theorem, by the way, the derivative at the point of minimum or maximum is equal to zero, well, which means that the tangential line is parallel to the x-axis, because derivative is a tangent of the tangential line with an x-axis. So, this is system of coordinates, then this line is parallel to this one, as well as this. So, now we will talk about a converse statement. If derivative is equal to zero, does it mean that it's either minimum or maximum? Well, the answer is no, but even if it does, question is which one is this minimum or maximum? So, that's what we're going to talk about. All right. So, first of all, let's just tell just a general statement, general definition, if you wish, a stationary point. Stationary point is a point where the derivative, first derivative is equal to zero. So, minimum and maximum are stationary points, as well as some others, which we will be talking about in the future. All right. So, first of all, let's just talk about, let's say, local minimum. Now, when I'm saying local, it means that the function might have something like this shape, in which case this is a local minimum, this is a local maximum, and this is also a local maximum. So, we are talking about local minimum at the moment, and what does it mean? Well, example, first of all, example, function f of x is equal to x square. At point, it's a parabola, as we know, and at this point, at point x is equal to zero, it has a minimum. So, it looks like a minimum, right? So, let's just talk about, from the perspective of derivatives. Now, let's just think about it this way. What does it mean that the function has a local minimum at this point? Well, it means that in the immediate neighborhood of this point, maybe small, but there is some kind of a relatively small neighborhood around this particular point, where the function is monotonically goes down, reaches the local minimum, and then monotonically goes up, right? Okay. Now, when the function is monotonically goes down, is decreasing, we know that its derivative is negative, because this is greater than 90 degrees, right? This angle is greater than 90 degrees, so the tangent of the tangential line is negative, and that's why derivative is negative. So, the function is, when it goes down, when it's decreasing, its derivative is negative. It's actually more precisely to say it's non-positive, because it can be equal to zero, at some particular point, if the function is really monotonically decreasing, and I will give you an example. So, let's talk about a derivative at this point to the left of the stationary point, in this case, its point where x is equal to zero. So, derivative is non-positive, so it's negative or zero. We know that for sure. Now, to the right of the stationary point, which is in this case a point of local minimum, function is increasing. When the function is increasing, its derivative, which is tangent of this angle, is positive or maybe zero, definitely not negative. So, on the left, we have non-positive, on the right, we have non-negative derivative. Now, let's talk about this particular point here. Now, this is the point where my derivative is equal to zero, because we're talking about this converse statement. If derivative is equal to zero, maybe it's minimum or maximum or whatever. So, we're talking only about these stationary points for derivative. First derivative is equal to zero. Now, to the left, I have non-positive, which means negative or zero. But it's not really zero, because we have zero here and we have determined that there is some kind of a neighborhood around it, where the function is really decreasing, right? So, to the left, we have basically negative. To the right, we have basically positive. So, the derivative is changing sign. If it's a minimum, it's changing sign from negative through zero at that particular point to positive. Now, what's the example? Example is x square. What's the derivative? It's 2x. Now, if x is negative to the left of the zero, this thing is negative. If x is positive to the right of the zero, this is positive. And at this particular point, x equals to zero, it's zero. So, it's definitely changing the sign from negative to positive as we go through this point zero. And this is the first and very important tool, which you can apply to differentiate between minimum and whatever is everything else, non-minimum. So, in case this is a minimum, if stationary point has this property of changing the derivative from negative to positive, changing the sign, if we go through this stationary point, that is a sign of having a minimum, local minimum at this particular point. And this is an example. And as you see, this derivative is changing sign from negative to positive as we go through zero. Now, let's talk about local maximum. So, let's consider the function minus x square. So, its graph would be this. So, that's local maximum, right? Now, here we have a very, very similar situation, but instead of changing the sign from minus to plus, in this case it's plus to minus. So, the function is supposed to be increasing in certain neighborhood. Again, we're talking about certain small neighborhood around this stationary point where no other stationary points exist. So, we're talking about the function about its stationary point and neighborhood where no other stationary points exist, which means our derivative equals to zero only at this point. Now, before that function is increasing, which means its derivative is positive or zero, non-negative, okay? Now, but it's not actually zero because zero is the only point where it's, where x is a stationary point. So, we can talk about positive derivative to the left of it. Now, to the right of the maximum function decreasing functions, monotonically decreasing functions have negative derivative or zero. But again, it's not zero if we are not at that particular point because this is the only point where derivative is equal to zero. So, it's negative here. So, positive on the left, negative on the right, zero is in the middle. That actually is, again, a sign of maximum, very similar to this one. So, if the derivative is changing the sign from positive to negative, that's the maximum, local maximum point. Well, let's talk about this particular derivative of this function is equal to minus 2x. For negative x to the left of zero, this is positive. And to the right of zero, when x is positive, this is negative. So, we have exactly the situation when sign is changing from positive to negative, which signifies the local maximum. Okay. And now, I would like to present the third case when the sign is not really changing. And here is an example. Let's consider the function x to the power of three. Here is the graph. Okay. Now, let's talk about this. Now, obviously, this is not minimum and not maximum, right? However, derivative at point zero 3x squared, that's the derivative. At point zero, it's equal to zero, right? If you put zero here, we will have zero. So, derivative is equal to zero, which means tangential line in this case is basically coinciding with the next axis. It's a stationary point. Therefore, x is equal to zero is a stationary point, because the first derivative is equal to zero. Is it minimum or maximum? No. And actually, there is a special name for this. It's called inflection, inflection point. It's when the function behaves like this. Now, let's analyze our first derivative. Now, in this case, function is monotonically increasing. And it's positive, right? Because zero is the only place where we have agreed some kind of a neighborhood where there are no other stationary points. So, function is monotonically increasing. And that's why its derivative is positive. Now, here, it continues to monotonically increasing. And again, the tangential lines will be always making an angle less than 90 degrees with an x-axis, which means the derivative is again positive. So, function is monotonically increasing. And its first derivative is always either positive or zero or positive again. So, there is no change in sign. It's not like minus to plus or plus to minus. It's plus through zero and then to plus again. So, this is a signification of non-minimum, more maximum, but being an inflection point. So, basically by analyzing how the derivative behaves around the stationary point, we have differentiated three different cases. One case when the sign is changing when we go through this stationary point, when the sign of the derivative is changing from minus to plus, another is from plus to minus. And the third one, it does not change the sign. So, it can be positive and positive or negative and negative. So, that third case is called an inflection. The first two are correspondingly minimum and maximum, whatever it is. I don't remember which one I said first. Okay, so that's the analysis of the behavior of the function using the first derivative. Now, let me make just yet another approach to the same problem. Okay, this approach is related to the second derivative. Now, let's consider the case when we have a local minimum. Now, the first derivative is 2x. Now, how the first derivative behave? Well, as we go through this zero point, it's first negative, then the negativity actually is decreasing, which means absolute value of this derivative is decreasing. And finally, it gets to zero. And then it goes to the positive and increasing, basically, from zero to the positive side. So, what I would like to say is that it looks like the first derivative is monotonically increasing from being negative through zero to being positive. Well, indeed, this function 2x is monotonically increasing, right? Remember the graph? It's just linear. That's the 2x. It's monotonically increasing. Now, what do we know about monotonically increasing functions? Well, its first derivative is supposed to be positive or zero, actually. So, non-negative. Well, let's just check in this particular case. So, the second derivative is the derivative of the first derivative is a constant 2. Now, it's positive, always positive, right? So, it looks like our second derivative is always positive, which means that the first derivative is always increasing in its value, right? It's monotonically increasing. Since the second derivative is positive, the first derivative is monotonically increasing, because the second derivative is the derivative of the first derivative, right? Now, if it's monotonically increasing and we know that it's zero at the stationary point, it means that before that, to the left of it, it was negative and to the right it was positive, which means it satisfies the condition of changing the sign, which means it's a local minimum. So, what I would like to say is that if the second derivative is positive at that particular point, it means that at this particular point, second derivative is changing the sign from minus to plus in this local minimum. Similarly, let's talk about minus x square. The first derivative is minus to x. The second derivative is minus to. It's negative. The graph is obviously this. It's local maximum. So, what do we see? That second derivative is negative, which means that the first derivative must monotonically decreasing. Since it's zero here, it means it's positive on the left and negative on the right. That's what's decreasing actually. And indeed, this tangent of this angle is positive, less positive, but less than. This case is zero and this case, the tangent is this angle. It's negative. It's greater than 90 degrees. So, here is the rule, basically. If the second derivative is positive, it's minimum. If it's negative, it's maximum. Well, let's talk about the function which has the inflection point. My first derivative is 3x square. My second derivative is 6x. Well, at point zero it's equal to zero. It's not positive and not negative, which means we cannot really apply this rule. Now, the function itself has really inflection point here, right? x cubed. So, basically it confirms that we don't have minimum or maximum. But it's not that simple. Sometimes, even if the function does have a minimum, let's say, if you cannot tell that the second derivative at that point, stationary point, is equal to positive or negative number. If it's equal to zero, if the second derivative is equal to zero, you still cannot say, using this particular rule, whether the function has minimum or maximum. And here is an example. Again, very simple example. Let's take the function x to the fourth. My first derivative is 4x to the third. My second derivative is 12x square. So, again, at point zero, now, first of all, the graph of this is, it looks like parabola, but it's much tighter to the x-axis at point zero. So, it's definitely minimum. There's no doubts about that with this function. x is equal to zero is definitely minimum. However, can we say anything using the second derivative at point zero? No, because it's equal to this. And that point is x is equal to zero. It's zero, which means we cannot use this rule. The rule, based on the second derivative at this point, can be used only if this particular second derivative at the stationary point is positive or negative, but not zero. Because if it's zero, we cannot say anything about the first derivative, whether it's going up or down or whatever. So, this is not always working as a rule. So, the second derivative rule is not always working. However, the first rule which I was talking about when the first derivative is changing the sign, in this case, this is working, because the first derivative is this, and on the negative x it's negative, and the positive x is positive. So, it does change the sign from negative to positive, which implies that this is a minimum. But this is based on analysis of the sign of the first derivative. Using the second rule, which I was talking about, the second derivative at the point of stationary point is not working in this particular case. So, it's a sufficient condition, but not necessary to have this, to have minimum if second derivative is positive, negative, etc. All right? So, we are not always using this rule. I mean, this rule might not always give us the result, the rule based on the second derivative. Rule based on the first derivative can do. But then you have to analyze, you have to really think about what is the neighborhood, you have to analyze, go to the left, within that neighborhood, go to the right. I mean, it's certain procedure. So, using the first rule, which checks the sign of the first derivative, whether it's changing or not, when you're crossing this stationary point, requires some a little bit more work, if you wish, or more analysis. Using the second derivative rule seems to be simpler. Just have the second derivative and plug the stationary point in it. But it's not necessarily working, because it might actually produce zero, which basically gives you nothing. All right. So, that's it for today. I would like you, actually, to recommend to read the notes for this lecture. And they also have some pictures. It's like a nice textbook. It's on this website. You can just go and read all the notes. And it's probably a good idea to do right now, just to refresh whatever we were talking about during this lecture. That's it. Thank you very much and good luck.