 maxima and minima, very interesting concept, this is the last application of AOD, maxima and minima. Now maxima and minima collectively is called extrema and you will come across two types of extrema, one is called local extrema, also called as relative extrema, also called as relative extrema. Extrema is a plural term, more than one extrema will make extrema. And the other type of extrema that we come across is called global extrema. Global extrema is also called as absolute extrema. So these are the two types of extrema, local extrema, global extrema. Now let us first talk about local extrema, global extrema I will talk about it later on in this course. Let us first talk about local extrema. Under local extrema we will talk about local maxima and local minima. Let me first define what is the meaning of a local maxima. Listen to this definition because this is the basic definition, of course calculus will help you to get it later on, but where calculus fails, this definition is going to save you. So you should be aware of what is the meaning of a local maxima, what is the basic definition of a local maxima. So a function is said to have, a function is said to have definition, a function is said to have local maxima at a point x equal to c, if it satisfies this condition, f of c should be higher than f of c minus h and in turn f of c is also higher than f of c plus h, where h is a very very small positive quantity. That means in the neighborhood of c the value of f c should be the greatest. By the way whatever we are studying here or whatever definition I have written over here that is a definition of strict local maxima. There is something called non-strict local maxima. What is non-strict local maxima? Non-strict local maxima which generally will not be discussed in your school exam or your school NCRT book. If your value at c is less than or greater than equal to what it has at c minus h and in turn is greater than f of c plus h. So here equality comes in strict local maxima only pure inequality is there. However we will be mostly talking about strict local maxima will not be talking about non-strict local maxima. To give you a simple example for this, a curve like this at point c it has got a local maxima. Now please do not take it as if it has to be differentiable there. I can have a point like this also where the function is not differentiable. Still there is a local maxima at this point. Or I could have a function like this also. Still it has a local maxima at this point. So this definition is not limited to any kind of continuous and differentiable function. You can apply to non-differentiable functions also. Or you can say continuous and non-differentiable also it can be applied and even discontinuous function it can also be applied. And you can see that in these two cases calculus will fail. So only in this case the calculus will work. So this definition is the one which is going to be useful even when the calculus fails to explain whether there is a local maxima at a point or not. This is a case of non-strict local maxima which is a case like this. Let's say if your function goes like this. What would you say for this point? Will you call this as a strict local maxima or a non-strict local maxima? The answer to this is it is an example of a non-strict local maxima because the value of c minus h is lesser than f of c but value of c plus h is equal to f of c. So what is happening here? f of c minus h is less than f of c. But f of c plus h is equal to or you can say equal to f of c plus h. So this will become a case of non-strict local maxima. Again you can see calculus will fail here because you will not be able to differentiate at this point c. However for my experience you will be mostly talking about strict local maxima only. So this is the one which is going to be more important for us. Is the basic definition of local maxima clear to you? Any question here? Sir we do one of the kind of local maxima, other one is kind of local minima. Real life examples of both. We have a full, I can say a segment of this chapter where you will be talking about maxima minima. Okay. I am sure you would have done practical applications. So I mean a typical example could be let's say how much amount of material will you use to maximize let's say if you have a fixed amount of material at your disposal what should you keep as the dimension of a cube for you to have a maxima volume? Okay. So here you are trying to find out the point of local maxima for that. So there is a full segment of application devoted to this. So we will come to that Venkat in some time. Notice in the next class we will come to it. Okay. Let's define local minima now. How is the local minima defined? Local minima definition. So a function, a function f of x is said to have, said to have local minima, local minima at x equal to c if f of c minus h has a higher value than f of c and f of c plus h also has a higher value than f of c. Again, just like the previous case it is called strict local minima. Okay. So what's a non strict local minima? It is where your f of c plus h sorry c minus greater than equal to f of c and this in turn is less than equal to f of c plus h. Okay. Some typical examples here I could cite a function which is behaving in this way. Okay. So this has got a local minima at c because c minus h and c plus h both have a higher value than f of c or it could be like this. Okay. Or it could be like this. An example of a non strict local minima could be like this. Got the point? Okay. But if I have a function like this, then can I say this is strict local maxima or non strict local sorry strict local minima or non strict local minima? It's a strict case. Yes. Because just before it the value is higher just after this the value is also higher. So it comes it comes under strict local maxima. So this is not the case here. This will be the case. Okay. Good. Now what is the role of calculus here? What is the role of calculus here? So first of all calculus can only work when your function is continuous and differentiable and when your function is continuous and non differentiable. So to a certain extent calculus can be applied only to those cases. So in order to study the position of local extrema by use of calculus will assume our function to be continuous continuous and differentiable. And we'll also talk about those cases where it is continuous but non differentiable and how does calculus help there? What is the role of calculus there? And there is a third case where a function is discontinuous. So there are three types of functions you will come across in your day to day problem solving one which is continuous and differentiable. That means there is no problem with the function whatsoever. Purely smooth. No breakage anywhere. Okay. That's where the calculus is of the greatest help. Second type of function that you'll see is continuous but is not differentiable at certain points. There also calculus can help you to a certain extent. Okay. And then there is a function which is discontinuous at certain points. There we'll see how do we, you know make use, how do you figure out local extrema things? Of course calculus is going to fail. So we'll see how we are going to tackle with those situations. So let me begin with the easiest situation. When f of x is continuous and differentiable, how do you figure out, how do you figure out where is the position of local maxima and where is the position of local minima? Okay. Differentiate. So yes, step number one is if you analyze both the maxima and minima positions or local maxima or local minima positions, you would realize that they always occur at such a point. They always occur at such a point where the derivative of the function vanishes. Okay. So if you see f dash c here is zero and f dash c here is also zero. Does it mean it is a sufficient condition for us to figure out the position of a local maxima or a local minima? No, you're not sure which one it is. No, let's say if I do f dash x equal to zero, will I always end up getting a local maxima or a local minima? Right. Now let me tell you the first step here is yes, you will differentiate it and you will put the function as zero and you will get the roots from it. Let's say one of the root is C. Okay. One of the roots is C. C need not be a maxima or minima. Are you getting my point? So as Thilpan is pointing out, it could be a point of inflection. So when I'm talking about continuous and differentiable function, you could end up getting a function like this, which is execute type function. You realize that here the slope will momentarily become zero. So let's say this was your x equal to C, f dash x equal to zero, but is it satisfying the basic definition of a local maxima or a local minima? I don't think so. f of c is higher than f of c minus h and in turn is lower than f of c plus h. That is not a definition of a local maxima. Local maxima definition is f of c minus s should be lesser than f of c and f of c should be greater than f of c plus h. That means it is the boss. f of c should have the highest value in the locality. It's the hero of the area. Okay. But in this case, even the derivative is zero. It is not a local maxima. Neither it is a local minima. Right? So let me tell you this is when you say f dash x is equal to zero and you get answers from it, all of them need not be your extrema points. Are you getting my point here? Correct? That is number one. Number two, even if you get your roots, let's say other roots are alpha and beta. Okay, how would you figure out which one is a local maxima and which one is a local minima or which one is neither? So first derivative test is just a step towards finding the extrema point, but it is not sufficient. It is necessary to do it, but it is not sufficient. Okay. So how would you figure out whether a point is satisfying this condition or it's the case one? Or it's a case two? Or it's a case three, which is an either case? How will you figure it out? Definitely. Will you give further insights? Aditi is saying second derivative test or number line test. Okay. Now let's say I want to make my decision on basis of only first derivative test. I don't want to go to any higher derivative test. I know some of you would be suggesting to do second derivative test. Okay. I do not want to do second derivative test. Tell me from first derivative test only I want to conclude whether the point is a maxima, minima or neither. How will I do that? f dash of c plus h will be negative. Okay. And f dash of c minus h will be positive for maxima. Okay. That's a very very good suggestion that Anjali has given. So when you draw f dash c, wavy curve. Okay. Of course, I'm showing it as three different number lines, but actually when you're drawing, you'll draw only one number line. And these points alpha c and beta will be drawn on that number line. If a point has to be a point of local maxima, you would realize that to the left of the curve, the derivative would be positive to the right. The derivative would be negative. That means f dash c minus would be a positive term and f dash c plus would be a negative term. Correct. Yes, I know. Now that can be easily be seen on the number line of the wavy curve. So make a wavy curve, put a sign, you'll come to see if there is a sign change from positive to negative. That means f dash c minus is negative here positive here and f dash c plus is negative over here. Suggesting that suggesting that at x equal to c, f of x has l max. Okay. So if a point has to have a local minima at a point, you would see to the left of the value c, the function f dash x would have a negative sign and to the right, it would have a positive sign. That means f dash c minus would be, sorry, would be negative and f dash c plus would be positive. Yes or no? If you see such as, you know, point which is showing this characteristic, you can conclude that at x equal to c, f of x has l min. But what is happening in those cases where it is neither or whether where it is not a local extrema, you see that either they will have both the signs positive or both the signs negative. Are you getting my point? In such case, in such case, your f dash c plus and f dash c minus will have same sign. If it is happening like this, that means the conclusion would be at x equal to c, f of x has neither. Now many people ask me, sir, can I call this neither as a point of inflection? Not so early. Yes, not so early. Why I'm saying not so early is something which I'll explain a little later on. At this step, we cannot conclude anything. If the signs both the directions are positive or negative, we cannot say it's an inflection. I'll give you one example to immediately conclude that. Okay, I'll talk about it later on. I don't want to confuse you there. Okay. Because inflection can have the derivatives till zero to the right and left of it. So I don't want to talk about it right now. I will throw more light on it when I do inflection in detail with you. As of now call it neither. Infection can be categorized under neither. Okay. Now, can I say from step number one and step number two, I will be able to figure out, I'll be able to figure out that a given point is a point of maxima, minima or a neither. Can I say this? No doubt about it. Now, doing this number line thing is not that convenient sometimes. Of course, when it is convenient, you should go for this approach. You should not try for higher order derivatives. Okay. Many times it is not convenient to do the second step because you need to make a number line for f dash x and you need to check the sign of the function f dash x in various intervals of x. That might not be very convenient at times. Okay. But let me go to higher versions of calculus only when I've done a simple question with you. So let me take a question with you and then we'll move on to the difficult and more in depth concept of maxima and minima. So let me take a small question. Okay. Okay. Okay. Okay. Okay. Okay. Let's do this question. Discuss. Discuss means find out the extremum of this function. Now, let me solve this question for you and then we can solve subsequent questions. Now, if you want to find out the maxima or minima of this function, you recall that numerator or you'll see here that numerator is a fixed quantity. The only varying quantity here is your denominator. Okay. So if you want to find out what is the maximum value of this function or what is the minimum value of this function and what are the points that they occur, then basically we have to focus on the denominator function. Okay. So let us call the denominator function as G. Let us differentiate this function. What do we get? 24 X square minus 36 X. Correct. Correct. Take a 12 common you end up getting in fact take a 12 X common. You end up getting X square plus 2 X minus three, which is actually factorizable as X plus three times X minus one. Okay. Now, let us make a number line and plot the zeros of G dash X on it. So minus three, zero and one. Let us put signs for this. So this will be positive. You all know your wavy curve very well. Positive, negative, positive, negative. Now, if you want to know where does your function G attain its maximum or minimum or whether there's a point, whether neither is happening, then basically you have to watch out for the signs on either side of a given number. For example, let's say I focus on a minus three. I see to the left of it G dash X is negative. That means the curve is coming down like this and to the right of it, it is positive. That means after momentarily stopping or after having a slope of zero, it start rising. The moment you make this rough graph, you'll come to know that this was a point of local minima for G of X. Remember for G of X. Okay. Right now I'm commenting on G. I'll come to F little later on. What can you comment about zero? To the left of it, it is positive. To the right of it, it is negative. That means the curve is having positive slope to the left of zero at zero. It becomes zero slope and after zero, it is having negative slope there by indicating that there is a local maxima occurring at zero. Are you getting my point here? Similarly at one, there is a negative slope on the left of it, zero at one and positive slope to the right of it, clearly indicating that there's a local minima occurring at one. There was no neither maxima nor minima case over here, so I'll not talk about it here. So what can you comment about F of X? If this is your analysis about G of X, what can you conclude about F of X? Can I say F of X will have local minima at X equal to zero because when your base is maximum, your entire function will be minima, correct? Yes, sir. And can I say it will have local maxima at X equal to one and X equal to minus three. Are you getting my point? So I never required a higher derivative test. Now, I do not call it as second derivative test like most of you would be calling it in school because that's a very half-azard knowledge that it is imparted in the school if somebody talks about second derivative test. It actually should be called as higher derivative test. We'll talk about that after the break. So I think I promised a break to you at 6.30, but it's already 6.45 now. So on the other side of the break, we'll talk about higher derivative tests when you don't want to make a number line. So let's have a break right now. Let me write here. Right now, the time as per my laptop is 6.47. I'm giving you some benefit of. Let's meet at 7.0, 7.0. Yes, sir. All right. Now let's talk about what is the alternative to step number two? What is the alternative to your step number two? That means if I don't want to draw the baby curve for f dash x, which probably may be difficult to draw or you don't want to draw the baby curve for f dash x, then what is the alternative for it? Now I'll come from the mistake that people make. The alternative for it is the second derivative test. Now when it comes to second derivative test, there are some false notions that people have in their mind. So what they do is they differentiate the function once more. Okay. And they try to put the point into the double derivative. Okay. If it comes out to be negative, then at x equal to c, f of x has a local maximum. There's nothing wrong in this. This is absolutely correct. Okay. If a double derivative at c gives you positive, then at x equal to c, your function has local minima. This is also absolutely correct. Why it is correct is because let us see the graphical analysis. If you have a local maxima at a point c, you would realize that the curve is concave downwards here. In other words, the slope value is going to decrease. As you can see in this entire neighborhood of c, in the neighborhood of c, the slope of the function is decreasing. So as to say that f dash x is a decreasing function in the neighborhood of c, where delta is a slightly positive quantity. So what will happen? It's double derivative should be, is double derivative should be negative in the neighborhood of c, in the neighborhood of c, which includes c point also. That's why f double dash c should be negative. Okay. And at the same time, the second test that you have written over here, that is also correct. I'm not denying that. What's the everything is correct? Then you were saying it is wrong and all. I'll come to the main point later on. So in the neighborhood of x equal to c, here you would realize that the slope is an increasing function. Slope of f of x is an increasing function. As you can see from negative slope, it is becoming zero and then it is becoming positive. That means f double dash x would be positive in the neighborhood of c. That means f double dash c should be also be positive. Okay. Now the mistake is the third point, which I'm going to write. When your f double dash c becomes zero, then people say at x equal to c, there is a point of inflection or there is a neither maxima nor minima. Correct? Is this what you people say point of inflection? By the way, what's the point of inflection? Let me define this for you. Point of inflection is a point. Point of inflection is a point. By the way, spelling of inflection is I N F L E C T I O N and there's one more spelling which is this both are fine. So what are the definition of point of inflection? It's a point where the curve changes its concavity. It's a point where your f of x changes its concavity. Concavity means from concave upwards, it can become concave downwards or for concave downwards, it can become concave upwards. Some examples would be like this. Okay. Here you see at this point, the curve is changing its concavity from concave downwards to concave upwards. It could be like this also. Okay. Correct? So from concave upwards, it is going concave downwards. It could be like this also. So this point concave downwards to concave upwards. Okay, it could be this point concave downwards to concave upwards. So this could also be a point of inflection. Okay. Anyways, I'll come to this little later on. But let me tell you this particular thing that you have written is not completely correct. How many of you believe that this was correct? So it could be stagnated also. Okay. We'll come to that. But this means double derivative test has failed. Remember, if your double derivative test, please, this is a very important point. If your double derivative test at a point C becomes zero, it just means double derivative test has failed. Please do not make any conclusion based on the double derivative test. That is what it tries to say. Okay. So that means we require higher order derivative test. We require higher order derivative test. So please do not make a judgment by the very fact that f double dash C is zero. It has to be a point of inflection. No, I'll give you a very classic example for it. Let's say I have a function x to the power four. Fine. I differentiated this function. I got four x cube. I put it to zero. And this gave me a point zero. Now I want to know what is the nature of this point? Is it a maxima? Is it a minima? Is it neither? Or is it a point of inflection? Okay. So when I do double derivative test at zero, you'd realize you'll end up getting 12 x square at x equal to zero, which is again a zero. Now from school point of view, this has, I mean not blaming school also. I'm blaming, you know, the limited exposure to functions in school that makes people believe that at x equal to zero, there would be a point of inflection. Okay. But in reality, at x equal to zero x to the power four, if you know the graph of x to the power four, this is the graph of x to the power four, I hope all of you know the graph of x to the power four. At x to the power four, this function has a minima. It has a local minima. So if you make your decision basis of this, that it is a point of inflection, you will end up concluding a minima as a point of inflection. Are you getting my point here? Yes, sir. Yeah. Now here, your second derivative test is getting failed. So in such cases, people ask me, sir, could we do that sign change thing? Yes, that would be a better idea to do it. So if you want to check the sign change, you'll see left to two zero, four x q will be negative right to zero, four x q will be positive. So there is a sign change from negative to positive, negative to positive. As you can see, there is a clear cut minima getting formed there. So sign change is a sure short way, but double derivative test here is not a sure short way because here it has failed. Okay. So people ask me, sir, what to do if it has failed? So yes, as I told you, it requires higher order derivative test, which we call as the nth derivative test. Okay. Now why did I say earlier that point of inflection when f dash x doesn't change its sign, it's a point of inflection. Okay. That is also not completely true is because when you look at a function like this, okay, let's have a function like this. To the left of this point, the slope is positive. To the right of this point, the slope is negative. So there is a change of the sign happening, but it is a point, this point C is a point of inflection. Right. So that is why I said it is not completely correct to say that. But here I am making an assumption that my function is continuous and differentiable. So that means this case cannot arise. So if this case cannot arise, then there would be no sign change happening in the neighborhood of a point of inflection. Are you getting my point here? See, this is deep concept. The way it is dead in school, it is not as simple as that. Okay. Is that clear? Yes, sir. Okay. So what is this nth derivative test? What does this test actually say? Let's talk about it in the next slide. So for a continuous and differentiable function, you can follow nth derivative test that is a sure-shot way to know whether a point is a maxima, minima, or a point of inflection. Now, what does this test say? Let me discuss it. The test says if you have a function, you differentiate it. Okay. And solve for this equation and you get one of the roots as alpha. And you want to know what is the nature of this point? Is it a maxima? Is it a minima? Or is it a point of inflection? Okay. So I'm assuming my function, again, I don't want to name it, but it's part of the same continuation. f of x is continuous and differentiable function. Okay. So now what do you do? You do the second derivative and you realize that you're still getting zero. That means second derivative test failed. You did third derivative and you realize that is also giving you zero. That means third derivative also failed. And you started doing this till you realize your derivative till n times at alpha is still giving you zero. That means nth, in fact, it is actually, let's say n minus 1, you realize it is giving you zero. Okay. But you realize that your nth derivative at alpha is not zero. Okay. Let's say this value is k. Let's say this value is k. Okay. Now, this is a flow chart that we normally follow. If your n is even, if your n is even, okay, and k is negative and k is negative, then at alpha, the function is having a local maxima. Okay. And if k is positive, then at alpha, your function is having a local minima. If your n is odd, if your n is odd, then at alpha, your function has a point of inflection, which can be categorized as a point of neither maxima nor minima or neither local maxima nor local minima. So when you learned your second derivative test, I think most of you do not remember the fact that if you go down for a minute, just a second, if your second derivative is zero and your third derivative is not zero, then only you can say at x equal to c, there is a point of inflection. f of x has a point of inflection. Many people, they do not care to check this. Okay. They miss this out. They miss out on this. Okay. That is what is covered up by your nth derivative test. So please make a note of it. So just by the... A little bit right, sir. Yeah. Just by the mere fact that double derivative at a point is zero, doesn't make it as a point of inflection. If double derivative is zero and triple derivative is not zero, then yes, it is a point of inflection. So if derivative at a given step is giving you zero means the test is insufficient, it has failed at that step. Thank you, sir. Sir, one minute. Yes, sir. Sir, could you please scroll left a bit? Sir, there is still nothing. Give us all the point of inflection, sir. Sorry. No, no. You still won't give us all the point of inflection. No, no. I'll talk about inflection points. They're not as simple to get. Yes. Done, sir. Okay. Now something about point of inflection. Something about point of inflection. Now, there is a debate in the field of mass about point of inflection. There are some authors who say that point of inflection should be points where the function is differentiable. But it is actually not true. Where I came across this was in one of the books that is normally taken by students for preparing for AP calculus, advanced basement calculus. Okay. I don't know exactly which one, but it was, I think the Princeton review book, which says that the function, the author of that book says that for a point of inflection, there must be a unique tangent at that point. But another famous book on AP calculus, which is Barron's, it says that it is not necessary. In fact, you can also check this out in your math stack exchange and all those websites where there is a forum for discussion on this. It says that it just says that point of inflection is that point where the curve changes its concavity. Okay. It may be differentiable there. It might not be differentiable there. Okay. Now I'll talk about point of inflection in such cases where your function is differentiable. Okay. For non-differentiability case, it is very difficult to detect such points. Now, if a function is differentiable at a given point. Okay. That means there is smoothness in the curve. How can you figure out a point of inflection? Okay. So point of inflection, as you can see from the previous nth derivative test, you can figure out the point of inflection from there. Okay. From graph, you can figure out point of inflection as that point. So point of inflection is that point where the tangent drawn to the curve also passes through the curve. I'm talking about differentiable curves. I'm not talking about non-differentiable cases right now because they're very difficult to detect. Okay. As I told you for non-differentiable case, you need to really go into higher versions of calculus to know whether there is a change in the concavity about that point. But for differentiable cases, it is pretty easy to check from the graph also. For example, let's say I have a function like this. Okay. Is there a point of inflection somewhere in between? You'll come to know the moment you draw a tangent there, you'll realize that the tangent crosses the curve at that point. Okay. I'll show you from GeoGebra curve. Okay. If we talk about x cube, if you talk about x cube, if I take a point here and draw a tangent at this point, you'll realize that the tangent is also crossing the curve at the very same point of tangency. Are you getting my point? So this is a point of inflection for us. Yes, tell me. So what do you mean by crossing the curve? Wherever it is a tangent, it is also cutting the curve at the same point. It is crossing the curve. It is cutting at the same very point. So cutting means like the value of the curve is higher at one side and lower at the other side? Not necessarily. See, it is just crossing the curve through that point. Yes, you can say that. Okay. I mean, the curve is to the either side of that line. Okay. Let me take another example. Let's say, let me draw a function for you. I can actually just give me a second. Just let me do some back work. Let's say I have a curve like this. Okay. I hope you can all see the green curve. I'll hide the x cube curve. Okay. Now, I'll show you something very interesting. I'll take a point on this curve. Let's say I take a point here. Let's say I call the point as a B point. Okay. I'll hide this point. At B, I will sketch a tangent. Okay. Tell me, is this tangent cutting the curve at the same point B? Yes or no? Is the tangent cutting the curve at the point B? No sir. No. Correct? The curve is on one side of the line. Isn't it? Now, let's start moving this point. Is it cutting now? No. No. No. No. Yeah. Of course, it is going and cutting at the other point. That is fine. But at the same point B, it is not cutting. Right? Now, see the moment I bring this down, down, down here, be very careful. Okay. Yeah. There you see. There you see. If you see at this point, you would realize that in fact, I'll take it a little bit more. Yeah. At this point, you would realize that your curve has started crossing the curve. Your, your tangent has started crossing the curve. That means now you're trying to go to a point where the tangent drawn is also cutting the curve at the same point. You see that? So this is a position where you would realize that the tangent drawn is also cutting the curve at the very same point. This is a point of inflection. So two things that get clarified from here is that one for point of inflection, it is not necessary that f dash x has to be zero. Right? It is not even necessary that f double dash x has to be zero. Okay. It is basically necessary that the tangent at that point must cross the curve at the same point. Are you getting my point here? Okay. So yes, tell me. Sir, I didn't quite understand what you meant by cutting, crossing the graph, sir, the curve. Is it some sort of intersection? Yeah, it is cutting through it. Cutting means what? It is passing through it, right? This line has been cut by this, this curve has been cut by this line, but this curve has been touched by this line. There's a difference. So I'll draw it. This is called touching. Orange line is touching the red curve, but this red line is cutting the red curve. That is the difference between cutting and touching. Touching means it will not pass through the curve at the same very point. It will just touch and go away. Okay, sir. But cutting means it will pass through it. So, sir, does that mean the curve and the tangent will have some area, some space in common, some line space in common? I didn't get that. What is the meaning of line space in common? Like, sir, in the black and the brown line, which I have drawn, from a certain portion is exactly coinciding, you can say. So could we say that? That's what I meant by common. It's a coincidence because of your human eye. If I zoom past it, it will not coincide. Can you draw e to the power x? It dies on the x-axis. It doesn't mean it starts coinciding with the x-axis at a certain point of time. It's the limitation of the human eye to detect that small change. It is not like they have a common segment together. If I keep zooming it, you'll definitely find a difference between it. Okay, see, if I draw a line like this, will you say it is cutting the curve at that point or will you say it is touching the curve at that point? Touching. But here it is cutting also at the very same point. It's a tangent, but it is also cutting the curve at the point of tangency. Yes. Yes, somebody was saying, excuse me, who was that? Here Ayush. Yes, sir. Okay. So the only way for such curves, which is continuous and differentiable, to know whether there is a point of inflection is in your first derivative test, in your first derivative test, or in your second derivative test, or in your third derivative test, whichever derivative test you're doing, if there is no, there's no change in the sign to the left and right of that point, okay? That means the derivative, so if you draw f dash x for a curve and you realize that there is no change in the sign happening, okay? That means at the point C, there is a point of inflection happening provided it is differentiable at that point. Okay. So if you take any slope, little right or little left to that point, you normally see the same sign of the slope. For example, x cubed graph, slightly right, you'll see a positive slope. Okay. Slightly left also you'll see a positive slope. So there's no change in the slope. Are you getting my point? That doesn't mean, that doesn't mean from this graph, many people have a wrong notion that point of inflection will always occur where the first derivative is zero. No. This is a, this is a classic example where f dash c is not a zero, but still there is a point of inflection. But still there is a point of inflection at C. Okay. So it is not very easy to detect the position of point of inflection. Are you getting my point? So if people think, yeah, if, if you people, many people think that if I have to find out all the points of inflection, I will just solve f dash x equal to zero and one of them will give me a point of inflection. No, that is not true. In this case, your f double dash x is zero, but f single dash x is not zero. f dash c is not zero, but f double dash c is zero. Okay. So it may happen that you have f double dash is not zero, f triple dash is equal to zero. Sorry, f double dash is equal to zero. If triple dash is not equal to zero, then there would be a point of inflection. So there are so many, you know, uh, uh, you know, tests that we need to perform in order to boil down onto a point of inflection, but thankfully we will not be going to such an extreme case. In fact, uh, the, the curves that are given to us, they are not a very high degree. They have a limited degree given to you. Of course, they will not give you a hundred degree term to figure out four or five or six degree maximum it will be. Okay. Or mostly you will be able to, you know, sketch the graph and come to know about it or you'll become to know from its point of inflection. But again, sorry, you'll be able to know from its single derivative test, but again, as I told you, you will not be able to get enough resources to figure out for a higher degree polynomial or very high degree polynomial to know where are the points of inflection. The best way is if you can sketch it, but that is also not very easy to do. Yes. Even by sketching here, how will we get to know sir? We can't exactly tell the point. Yeah. Sketching means you have to do some graphical testing on it. Okay. For example, the nth derivative test will give the absolute solution of all of the example, if you're to curve like this, you know that there is a point of inflection that will occur somewhere over here. Right? You know, there's a point of inflection that will occur somewhere over here. You know, there's a point of inflection that will occur somewhere over here. Okay. So from rough positioning of the graph you can make out. Yes. Because as you can see from your nth derivative test, let me go back from your nth derivative test, only when your f nth derivative is not equal to zero and n happens to be an odd number. Okay. Then there could be an inflection point at alpha. Then there can be an inflection point at alpha. So you have to go till that extent. But this has been f dash alpha is zero. Huh. If you start getting all of them as zero. Yes. That's what this also gives you some limited version of the points. You miss out on those inflection points where f dash x itself was not zero and there was a point of inflection. That's exactly the graph which I've drawn for you. Yes. Okay. So yes. This is only when your f dash x is known to be zero. You can go and find out whether it was a point of inflection or not, but not for other point of inflections will be missed out in this case. Okay. So let's take up some questions on whatever we have studied so far. Sir, can you please show that graph page, the one which you wrote about the graph. The black page. The black page. The previous page, right? Yes, sir. This page? Yes, sir. Sir, just a minute, please. Yeah, sure. Yeah. Find the extrema. Find the extrema for the function x minus two to the power four times x plus one cube. Yes, done. Anybody who's done with it? Okay. Very good. Okay. Let's discuss this. First thing that I would do is find f dash x. Now, if you take x plus two to the power three and x plus one to the power two common, you'll end up getting four x plus one plus three x minus two. So this will give you x minus two cube x plus one whole square. And this will give you seven x plus seven x minus two minus two. Now, the zeros of f dash x equal to zero, you plot it on the number line. So minus one, two by seven and two. Let us put the sine scheme. This is positive. And now when you're crossing two, two comes from a factor which has got odd power. So there'll be a switch of sine and then two by seven comes from again a factor which has got odd power, which will again mean a switch of sine. And minus one comes from a factor which has got even power. That means there's a retention of sine. Okay. Now, from here, we can conclude number one, that minus one, there is no sine change to the left of it. That means it's a point of inflection. But does it mean it is the only point of inflection? No, there can be more points of inflection. Are you getting my point? This is a point of inflection. But that is that doesn't mean it is the only point of inflection. Next, two by seven, there is a change of slope from positive to negative. So before it, the curve was having positive slope. After this, the curve is having negative slope, which clearly means it's a point of local maximum left to it negative slope, right to it positive slope. That means it's a point of local minimum. Okay, so roughly it gives you the nature of the curve that it would be like this at minus one, it will be like this momentarily, it will be stopping. Okay, then go high at two by seven, come down at two, like this. Now, when it is happening like this, let me tell you, there could be an inflection point here, right? There could be an inflection point here, which is not caught here, which we are not able to detect here. Let us draw the curve on geojibra and check. Somebody please dictate the function to me. What was it? Y is equal to X minus two to the power four. X minus two to the power four. X plus one whole cube. Yeah, there you go. Do you see at minus one, there is a momentarily becoming slope zero. So it was a point of inflection, right? So it went and achieved its maximum somewhere on top, which is at two by seven. Okay, now when it is coming down, it achieved its minima at two. But somewhere in between, there would be a point of inflection. Somewhere here, there would be a point of inflection. Okay, so this is one point of inflection, but it doesn't mean that is the only point of inflection. There can be more points of inflection. Let us talk about this question. I'll put the poll on. I think next class also will continue with maxima and minima. I have to talk about those functions which are non-differentiable, which have got corners and cuffs. And I'll also talk about discontinuous functions, how to figure out where is the point of maxima and where is the point of minima for those functions. This time I've given you a function in a parametric form rather than giving you in a Cartesian form. Okay, one person has responded so far. Okay, last one minute. Okay, five, four, three, two, one. Okay, quite a mixed response. A and C has got equal number of votes. Okay, now this question is a very big eye-opener for most of you, I'm sure. Let's solve this question. Now, when you're trying to find out what is the point of maxima and minima, we normally find out the derivative of the function dy by dx. So for a parametrically defined curve, it will be dy by dt. Yeah, so dy by dt divided by dx by dt. Okay, so dy by dt would be 12t square minus 60 minus 18, right, divided by 5t to the power 4 minus 15t square minus 20. Okay. Now, I think this is factorizable. If you take a factor of 6 out, you get 2t square minus t minus 3. And this is 5t4 minus 3t square minus 4. Okay, this again, I think is factorizable. It is 2t minus 3 times t plus 1. And this is going to be 5t square plus 1 into t square minus 4, right. t square minus 4, you can write t minus 2t plus 2. Okay. Now, see, many people, you would have done this. You would have made a wavy curve for dy by dx. So the least point is minus 2, then you have a minus 1, then you have a 3 by 2, and then you have a 2, correct? Am I right? Sine also you would have put, this would be plus, okay. This would be minus plus minus. And I don't have to go any further. And remember, t is only between minus 2 and 2. So I don't even have to worry about this. So I'm only looking in this interval from minus 2 to 2. Now, looking at this sign, many people will conclude that there is a change of slope from negative to positive, okay, at minus 1, correct? Yes or no? And change of positive to negative at 3, at 3 by 2, correct? So this makes people conclude that at minus 1, there is a minimum, correct? And I think that is the main reason why many of you also have voted for a option. But you'll be surprised to know that at minus 1, actually there is a maxima, okay? Let us do double derivative test. Okay, what is double derivative test for this? You have to differentiate this function of t again, okay? You have to differentiate this function of t again, okay? So let me write down the double derivative. So 6 by 5 is a constant. So I'll just, let me just drop it down. Yeah, 6 by 5 is a constant. And you'll end up getting t to the power 4 minus 3t square minus 4 into derivative of this term, which is 4t minus 1 minus 2t square minus t minus 3 times 4t cube minus 60 whole divided by t to the power 4 minus 3t square minus 4 whole square. Now remember this is derivative of, see, you're doing this d by dx of a function of t you are doing, right? Correct? So you have to do like this d by dt of f of t into dt by dx, correct? So dt by dx also you have to divide over here, which most people forget. So you have to divide by t to the power 4 minus 3t square minus 4, okay? Now here, if you put your t as minus 1, see what will happen? Put t as minus 1 in this, what will happen? Check. So 6 by 5, this will become 1, this will become minus 3, minus 4. This will become minus 5, correct? Minus 5. What will happen here? 2 plus 1 minus 3, 2 plus 1 minus 3. Okay, that will be 0. So anyways, I don't have to worry about this term over here. So that will be 0 anyhow. Now this is a positive term, so I don't have to worry. And this will be a 1 minus 3 minus 4, which is a negative term, correct? So see what has happened? 6 by 5, negative, negative, positive. Denometer also positive, but this is negative. So overall, overall this answer has come out to be negative. What does it mean? At minus 1, there is a maxima. Now can somebody explain me why this number line test failed over it? So because it's in respect to t and not t. Right. When you're doing such judgment, it is actually absurd to do this judgment. It is not a real x-axis, it is at axis of t. t is itself defining your x in some different way. See, when you're trying to see the slope of the function, you see what is the rise of dy by dx with respect to x? What is the slope with respect to x? You don't see a slope with respect to the parameter t. This is a mistake which I have seen people doing. They start applying the number line test when the function is parametrically expressed. So please do not do this. This will give you wrong results. I purposely made you do this so that you appreciate that what mistakes you may do in the exam. Are you getting my point here? So people who have said this will definitely get a minus 1. It is not the right answer. Okay. So the other point zero and half are not even in the game. So zero and half need not be entertained. So only point left is 3 by 2. So you can go for a 3 by 2 or check for yourself that at 3 by 2, this will give you a positive value. Okay. So please do that check. I will not do that check for you. So confirm that d2y by dx square at 3 by 2 is positive. Okay. So option d is correct in this case. Are you getting my point here? So what did you learn from this? What did you learn from this particular question? Never apply the wavey curve sine scheme to know whether a point is a point of maxima or minima when the function is parametrically defined. Of course if t is equal to x, then probably you can say that. Oh excuse me sir. Yes, sir. But here in the answers, they are asking for the value of t itself, right? So they are asking you the value of t. Sorry? Understand the value of t. We can't use the wavey curve sine scheme. They are asking you the value of t. Of course. But it doesn't mean that at those values is a maxima and minima. That is not very evident from the wavey curve made on t. See, when you have a function like this, right? And the graph is like this. Correct? It is, you are able to see because you are drawing it, you are drawing y with respect to x. Correct? So you can say that, okay, let's say this is your, you know, point alpha, this is point beta and this is point gamma, let's say, okay. Then you can say from here to here, the function is increasing. So this will be having a positive sign here. Here to where the function is decreasing, right? Because your function can be seen with respect to x. But you can't comment positive and negative on t because t is not your x-axis. Your x-axis is itself dependent on t. It is a function of t. So we can't judge the increasing, decreasing values of the slope or increasing, decreasing nature of the function by seeing its variation of dy by dx on t-axis. No, it is not the t-axis. You have to see it on x-axis. That's why the wavey curve line scheme failed there. Okay. So we'll stop here. Next class when we meet, we'll be talking about, as I told you, application of maxima and minima. We'll talk about finding the maxima and minima for functions which are non-differentiable or for functions which are discontinuous. We'll also talk about global maxima and minima. So I think initial two, two and a half hours of class will go into that. And then last one hour, we can introduce definite integers. So all the best.