AI Assistant
Transcript
47:41:730Paolo Guiotto: F or CB level?
47:46:30Paolo Guiotto: I'm sorry, but unfortunately, it's quite crazy. I don't know how…
49:05:80Paolo Guiotto: Okay, so now let's see another problem here. The main difference is that the domain is not compact, and there is no boundary, okay? So this is what is usually called a free optimization, because variable
49:22:680Paolo Guiotto: variables, X and Y, can vary in the interior line, so point XY can vary in the full plane R2.
49:32:470Paolo Guiotto: However, let's, let's see what this problem asks. So, first question, we have to compute the limit of this function. It will have a relevance in minimum-maximum problem. So,
49:48:860Paolo Guiotto: What about the limit? That's a problem of computing a limit at infinity of this function. We may notice that, for example, if we look at the X section.
49:57:930Paolo Guiotto: We put y equals 0, we get X to power 4. That goes to plus infinity, no matter how X goes to plus minus infinity, which means a modulus of X goes to plus infinity.
50:12:530Paolo Guiotto: Which means, in this case, modules of X corresponds to the norm of X0.
50:18:700Paolo Guiotto: So when point X0 goes to the infinity, okay? So that's equivalent of saying X0 Goes to infinity.
50:28:290Paolo Guiotto: Now, is that sufficient to conclude anything?
50:36:500Paolo Guiotto: Can I say that the limit of F is infinity?
50:52:900Paolo Guiotto: So, what should we do?
50:56:860Paolo Guiotto: Yeah, so…
50:58:590Paolo Guiotto: Sections are never sufficient to conclude about the existence of the limit. They are useful to disprove existence if you find two different sections along which you have two different limits. At this stage, I can only say that if
51:15:490Paolo Guiotto: A limit exists.
51:17:620Paolo Guiotto: If limit… at infinity.
51:21:280Paolo Guiotto: of F exist, It must be… equal to plus infinity. But…
51:32:850Paolo Guiotto: I cannot yet say that this is true.
51:53:370Paolo Guiotto: Okay, so to check this, if we bet on the fact that the limit will be plus infinity, let's see what is the function in polar coordinates. So, if I put X equal raw cosine
52:08:180Paolo Guiotto: Theta, Y equal raw sine theta.
52:12:680Paolo Guiotto: Now, the good thing is that X squared plus Y squared is raw square, so this becomes raw square to power 2 minus
52:24:420Paolo Guiotto: Then we have 3X square Y, so 3, x squared is raw square cos square theta, Y is raw sine theta.
52:34:570Paolo Guiotto: So I put together these two raw, and I have a raw power 4 here.
52:41:70Paolo Guiotto: Minus 3, raw cube.
52:45:860Paolo Guiotto: Then we have another coefficient, which is a non-constant, cos square theta sine theta.
52:52:300Paolo Guiotto: However, if I look at this, remind that point XY goes to the infinity
53:00:320Paolo Guiotto: means that the norm of XY goes to plus infinity. And the norm of XY is exactly raw.
53:11:860Paolo Guiotto: So, it means raw goes to plus infinity, and about theta, you know only that theta is in the interval, 0 to pi.
53:21:130Paolo Guiotto: But you cannot say anything about this theta, anything more than this.
53:25:610Paolo Guiotto: So, what we see here is that we see that if we look at this as a function of rho, it is clear that this function goes to infinity, because rho power 4 dominates rho cube. The unique problem is that we have a…
53:39:170Paolo Guiotto: And on constant coefficient.
53:41:910Paolo Guiotto: So we have somehow to eliminate the dependence on theta, and the right thing to do if we bet on the fact that it goes to plus infinity is to bound from below, okay? So I want to say that this is greater or equal than
53:56:350Paolo Guiotto: Ra power 4 remains, I can do… I don't need to do anything, and since this quantity.
54:03:70Paolo Guiotto: No? I am something like subtracting 3 times raw cube times this coefficient. How big can be this coefficient?
54:11:250Paolo Guiotto: It can't be bigger than one.
54:14:80Paolo Guiotto: So, at most, I subtract 3 raw cubes, so this is greater than this.
54:20:990Paolo Guiotto: Which is good, because when rho goes to plus infinity, it goes to plus infinity as well.
54:26:890Paolo Guiotto: So this now says that there exists the limit
54:32:250Paolo Guiotto: When XY goes to infinity in plane.
54:37:640Paolo Guiotto: of F, and that limit is equal to plus infinity.
54:41:550Paolo Guiotto: Okay? So question 2. We have to determine these stationary points of this F.
54:53:890Paolo Guiotto: We're not…
54:58:450Paolo Guiotto: square plus Y squared squared minus 3X square Y.
55:04:480Paolo Guiotto: So clearly, it's a continuous function, it is a polynomial. The derivative with respect to X is equal to 2X square plus Y squared times the derivative of this argument with respect to X, which is 2X, minus 6XY.
55:22:670Paolo Guiotto: And the derivative with respect to Y is similar, so we have still 2x squared plus Y squared times derivative of the argument, now with respect to Y, so it's 2Y, minus 3X squared. As you can see, they are both continuous functions on R2.
55:42:640Paolo Guiotto: So this says that the differentiability test applies
55:49:220Paolo Guiotto: It says that this function f is
55:53:420Paolo Guiotto: Well, it's write in red. F is… differentiable, on R2.
56:01:880Paolo Guiotto: Okay?
56:03:130Paolo Guiotto: So this is just an information. Now, XY… Ease.
56:09:50Paolo Guiotto: stationary point.
56:11:750Paolo Guiotto: 4F…
56:13:540Paolo Guiotto: If and not if the gradient of F at point XY is 0, 0 vector 0. And this means that the two partial derivatives must be equal to zero.
56:25:940Paolo Guiotto: So, the first one is, 4X… well…
56:30:790Paolo Guiotto: I wouldn't suggest you to, for example, to develop factors, because the best thing, since this is going to be an equation, equals zero, is to factorize.
56:41:770Paolo Guiotto: Because when you have a product equals zero, it means at least one of the factors is zero. So, here, for example, you see that there is a common factor, which is this X,
56:53:320Paolo Guiotto: And I should factorize this. If you want, we can factorize also a 2, for example, so we keep a 2X, and then we have, parentheses, 2 times X squared plus Y squared.
57:08:770Paolo Guiotto: Now, minus 3Y, all this equals 0.
57:16:90Paolo Guiotto: But the second line, there is nothing that can be factorized, so we have to…
57:21:680Paolo Guiotto: keep this equation, maybe we write just 4Y times X squared plus Y squared minus 3X squared equals 0.
57:34:550Paolo Guiotto: Now, as you can see, this system is not immediately evident, because there are complicated equations.
57:41:70Paolo Guiotto: I would say we should start from the first one.
57:45:30Paolo Guiotto: Because at least here we have a little alternative that reduces a bit the complexity. From the first one, we can say that product, part that this can be canceled, that product is zero if and all if.
57:59:160Paolo Guiotto: Either X is 0, or the bracket is 0. So we have two sub-cases. X equals 0, or the bracket equals 0. This means 2X squared plus Y squared minus 3Y is equal to 0.
58:16:70Paolo Guiotto: Now, let's see what happens in the second equation in these two circumstances. In the second equation, when we plug x equals 0, you see that there is a lot of… there are lots of terms that disappear, and actually, it remains only 4Y cubed.
58:33:190Paolo Guiotto: If I'm not wrong, equals zero, which is something that can be easily solved, because this means Y equals zero, no?
58:41:590Paolo Guiotto: So, at the end, the first alternative yields X equals 0, Y equals 0. So, the solution is 0, 0 as a point.
58:51:650Paolo Guiotto: About this cycle, what could we do?
58:55:990Paolo Guiotto: What could we do here?
59:02:120Paolo Guiotto: You don't see anything.
59:14:610Paolo Guiotto: I don't like.
59:16:640Paolo Guiotto: No, sorry, can you repeat?
59:26:80Paolo Guiotto: So we want to say, let's exploit this, right?
59:29:870Paolo Guiotto: Say, make explicit this and plug into the second. Yeah, that's the good idea, okay?
59:35:570Paolo Guiotto: So, because you have the same element here, no? So you could say, okay, the second equation remains, so I do not write, and now I do this here. I say X squared plus Y squared
59:49:110Paolo Guiotto: Be always careful to… when you divide, that you are not dividing by a variable. That could be zero. Otherwise, you should just say, if it is zero, this happens. If it is different from zero, these cells up.
00:00:910Paolo Guiotto: Now, this is 3 half Y, and I plug this into the second line, no? Here. So I get 4y times that X squared plus Y squared, which is 3 half
00:15:60Paolo Guiotto: Y minus 3X square is equal to 0. And as you will see, this will be much better, because now I continue just with this second, the first one is over, no? So I have the first equation is X squared plus Y squared equal 3 half Y.
00:33:900Paolo Guiotto: This one, let's simplify a bit. We can, for example, we can divide by 3, which is different from 0. Throw away, this is, 2.
00:43:300Paolo Guiotto: So we have 2Y square minus X squared, which is not immediately, maybe, evident, but it's something… if you have a Y square minus X squared, you would do what?
00:55:270Paolo Guiotto: If that is not de facto 2, you would say that this is
01:00:940Paolo Guiotto: Y minus X times Y plus X. You can do the same here. You can just say it is root of 2Y minus X times root of 2Y plus X equals 0.
01:13:730Paolo Guiotto: Okay? Because in this way, I have, again, a product. This says that I have two sub-cases. One is the case root of 2y minus X equals 0, or the other is root of 2Y plus X equals 0.
01:31:320Paolo Guiotto: So this is going to be Y equals some function of X, EZ function.
01:36:640Paolo Guiotto: Or you can send me better, X equals a function of Y, so let's… the second line is still X squared plus Y squared equal 3 half Y, so X squared plus Y squared equal 3 half Y.
01:54:610Paolo Guiotto: So perhaps it's better if I say X equal, so if and only if. The first one is X equal root of 2Y. I plug this into the second line. I have to continue to do these kind of things until I finish, until I find a solution. When I plug this into the second, I get 2Y square plus Y squared. You see, this goes here.
02:18:240Paolo Guiotto: No?
02:20:00Paolo Guiotto: No, it's this X squared, equal 3 halves Y, and similarly here, I have X equal minus root of 2Y,
02:29:770Paolo Guiotto: But when you replace in the second, the same equation comes out, so 2Y squared plus Y squared equal 3 half Y. The nice thing is that this is a simple equation in Y,
02:42:180Paolo Guiotto: So let's simplify and write, be careful, I'm sure that you would divide by Y.
02:48:290Paolo Guiotto: But to divide by Y, you have to say, if Y is different from 0, I can't do that. Otherwise, I can't do that, okay? So, it's better…
02:57:380Paolo Guiotto: to do this. So, for example, this is 3, as you can see, this 3Y square. We can cancel this 3.
03:06:930Paolo Guiotto: If you want, we can carry the 2 on the other side. We put the Y on the left-hand side, so we have 2Y squared minus…
03:16:950Paolo Guiotto: 2Y square minus y equals 0, which is a second-degree equation. You don't need to use formulas, because factorizing Y, you read Y times 2Y minus 1 equals 0.
03:29:400Paolo Guiotto: And since, you see that is exactly the same of this, we get the same equation. So here we have X equals minus root of 2Y, and then Y times 2y minus 1 equals 0.
03:44:630Paolo Guiotto: We are almost at the end. So, now, this is two sub-cases that comes now from this equation. We have either Y is 0, or 2Y minus 1 is 0, y equals 1 half.
04:00:280Paolo Guiotto: Same here, Y equals 0, or Y equals 1 half.
04:07:160Paolo Guiotto: Now, X is root of 2Y for the first two alternatives, so this is X equals 0, this is X equals root of 2
04:17:110Paolo Guiotto: Times 1 half, so root of 2 over 2, or if you prefer, 1 over root of 2.
04:22:400Paolo Guiotto: Here, we have X equals minus root of 2Y, so when y is 0, X is 0. When Y is 1 half, x is minus 1 over root of 2.
04:32:710Paolo Guiotto: Now the solution is complete, and let's see what are solutions. So, do not miss this one.
04:40:550Paolo Guiotto: That, we found before, which is again found here, and here it is always the same solution. So, the conclusion is that stationary points
04:52:430Paolo Guiotto: R.
04:53:810Paolo Guiotto: 0, 0.
04:55:960Paolo Guiotto: Then we have one half 1 over root of 2,
05:01:130Paolo Guiotto: and 1… oh, no, no, sorry, sorry, sorry, sorry. I inverted. X is 1 over root of 2.
05:09:260Paolo Guiotto: Y is 1 half. Here, X is minus 1 over root of 2, and Y is 1 half.
05:16:450Paolo Guiotto: And these are the stationary points. So we found 3 stationary points.
05:21:870Paolo Guiotto: Now we have to discuss the minimum-maximum problem, which is, let's say, the
05:26:880Paolo Guiotto: These are auxiliary questions that will be useful to discuss minimum-maximum. First, we notice that the domain is not compact, because it is not bounded, so here, Weiss's theorem directly does not apply.
05:43:740Paolo Guiotto: However, we mentioned that,
05:47:180Paolo Guiotto: When we are on closed and unbounded set, and we know the behavior at infinity of the function.
05:54:190Paolo Guiotto: We can say if there is one of the two, no? Remind this.
05:59:240Paolo Guiotto: So, since perhaps you forgot this, let's remind. Let's… reminder… this… Important.
06:15:230Paolo Guiotto: Fact.
06:16:810Paolo Guiotto: which is actually a corollary of Weisser's theorem.
06:20:620Paolo Guiotto: That if you have a function F, which is continuous on D, where D is closed, but unbounded.
06:35:480Paolo Guiotto: Which is our case, because the full space is closed.
06:39:260Paolo Guiotto: and unbounded.
06:41:960Paolo Guiotto: And you know that, for example, the limit, as in this case, when X goes to the infinity of the space, this holds in any dimension, of f of x is equal to plus infinity.
06:56:760Paolo Guiotto: So the idea is that you imagine, no, to have an idea that this is RD, so R, no? And the function blows up at plus minus infinity.
07:07:690Paolo Guiotto: So, you may imagine that if the function is continuous, it will do something, but somewhere in the middle, it will have a minimum. So that's what theorem says. Then, there exists a global minimum.
07:21:390Paolo Guiotto: for F on D.
07:23:580Paolo Guiotto: And, of course, there cannot be maximum, because if the function has limit equal to plus infinity, it means that the values of F can be made big as you want, no? Provided you take X sufficiently far from the origin. So there cannot be a value where f is bigger than all the other values.
07:44:480Paolo Guiotto: and… Of course.
07:50:340Paolo Guiotto: There is no maximum.
07:53:580Paolo Guiotto: for F on the…
07:56:430Paolo Guiotto: There is also a dual statement that says if you have the limit equal to minus infinity, so the function goes down, there will be a maximum somewhere, there is no minimum.
08:06:360Paolo Guiotto: So that's exactly what we have in this case, because we have a domain, which is R2, closed and unbounded, a function that goes to plus infinity at infinity, so we are exactly in the configuration, and the function is, of course, continuous.
08:22:779Paolo Guiotto: So, in our case.
08:28:420Paolo Guiotto: We have… F, which is continuous on R2, R2 is, closed.
08:44:779Paolo Guiotto: and unbounded.
08:49:880Paolo Guiotto: And moreover, we have also verified, at first question, that the limit for XY going to infinity
08:59:770Paolo Guiotto: In plane is equal to plus infinity.
09:03:680Paolo Guiotto: So… We say that there exists a minimum, for F, on R2… And, there is not maximum.
09:16:160Paolo Guiotto: 4F.
09:17:330Paolo Guiotto: In R2.
09:18:569Paolo Guiotto: So let's focus on minimum now.
09:21:840Paolo Guiotto: How can we determine? This is a… now, we know that this point exists.
09:26:529Paolo Guiotto: And now we can say, if it is in the interior of the domain, it must be a stationary point, because the function, we already discussed this, is differentiable. So Fermat theorem applies.
09:38:660Paolo Guiotto: Otherwise, if it is in the boundary, we don't know. But what can be said in this case? Is there this alternative?
09:52:430Paolo Guiotto: What is the interior of D in this case?
09:56:620Paolo Guiotto: Since… interior of R2 is… What is the set of points which are contained into R2 with the ball?
10:08:110Paolo Guiotto: It's R2 itself, R2 is open, if you want.
10:12:40Paolo Guiotto: I have two E's.
10:14:370Paolo Guiotto: Open.
10:16:440Paolo Guiotto: It's open and closed, is basically with the empty set is the unique exception to this.
10:23:280Paolo Guiotto: So we have a unique alternative. If, X or Y?
10:30:150Paolo Guiotto: is a minimum.
10:34:240Paolo Guiotto: for F, being… differentiable.
10:44:930Paolo Guiotto: Ferment.
10:48:550Paolo Guiotto: theorem.
10:50:550Paolo Guiotto: applies…
10:54:10Paolo Guiotto: And so we conclude that XY must be necessarily a stationary point. There is not the other case, there is no boundary, no? XY
11:05:280Paolo Guiotto: is stationary point.
11:09:350Paolo Guiotto: for F.
11:11:220Paolo Guiotto: But, we already found the stationary points. We discovered that there are 3 stationary points, so it means that minimum is one of these three points.
11:20:560Paolo Guiotto: So, minimum.
11:23:840Paolo Guiotto: point.
11:25:230Paolo Guiotto: is 1… Or more, why not? There can be two, three minimums.
11:31:350Paolo Guiotto: Or more.
11:35:110Paolo Guiotto: off… this. 0, 0.
11:39:60Paolo Guiotto: I do not remind, we have plus minus 1 over root of 2, 1 half.
11:47:510Paolo Guiotto: Now, you see that the search is restricted to 3 points.
11:51:580Paolo Guiotto: So, the value here, looking at the function F,
11:59:620Paolo Guiotto: The function is this one. When you put the X and Y equals 0, get 0, clearly. So here you have 0.
12:07:220Paolo Guiotto: About this, you notice that, since the function
12:13:10Paolo Guiotto: depends on X. Through X squared, you see this
12:18:590Paolo Guiotto: And this. So plus minus is the same, no? When you put plus something minus something, you get the same value. So the value at these two water points is the same.
12:30:620Paolo Guiotto: and it is equal to… so here, we have, F equal to, so…
12:39:960Paolo Guiotto: We have to do the square of the first, which is 1 half, plus the square of the second, 1 fourth to power 2,
12:49:540Paolo Guiotto: So that's X squared plus Y squared.
12:52:640Paolo Guiotto: Minus, there is 3X square Y, right?
12:57:50Paolo Guiotto: So 3, X squared is still 1 half, Y is 1 half.
13:03:80Paolo Guiotto: So, at the end, we have,
13:05:510Paolo Guiotto: This is, 3 fourths square, so 9 over 16 minus 3 over 4.
13:14:380Paolo Guiotto: This means that we have,
13:20:740Paolo Guiotto: Minus 12, right?
13:23:90Paolo Guiotto: No, no, no, no.
13:24:760Paolo Guiotto: Minus 3 is minus 12 at the other. So 9 minus 12 divided 16, so minus 3 over 16.
13:33:710Paolo Guiotto: So, the value here is 0, the value here is minus 3 over 16, which is less than 0, so these are the minimum points.
13:42:640Paolo Guiotto: So, conclusion…
13:47:750Paolo Guiotto: mean… points.
13:50:840Paolo Guiotto: R.
13:52:700Paolo Guiotto: So these two, plus, minus 1 over root of 2, One half.
14:00:350Paolo Guiotto: Okay?
14:05:370Paolo Guiotto: Okay, so let's do the next one, which is a problem…
14:11:670Paolo Guiotto: A little bit more complicated, because we have 3 variables, but the philosophy is the same.
14:18:780Paolo Guiotto: So, exercise…
14:24:560Paolo Guiotto: 2912.
14:26:920Paolo Guiotto: Is there any problem?
14:28:700Paolo Guiotto: Okay.
14:31:480Paolo Guiotto: There is nothing to laugh, okay?
14:35:740Paolo Guiotto: Oh, no, why?
14:39:990Paolo Guiotto: You're a mazogist That's what I say here.
14:45:650Paolo Guiotto: No, but I mean, this is not real world. Real-world problems are made of 50 variables, or 100 variables, you see? So…
14:57:870Paolo Guiotto: It's much harder, and you, of course, won't solve explicitly. But you have to understand the main ideas, because if you, for example, you write an algorithm, you use a code, you know, a software, you must know what to do, okay?
15:21:630Paolo Guiotto: So, we have this… It's similar to the previous one. 4. XYZ, in R3.
15:34:230Paolo Guiotto: Same, basically, not basically, same questions. So, number one, limit at infinity of F, number two, stationary points.
15:46:300Paolo Guiotto: Number three, mean max…
15:53:640Paolo Guiotto: of F on R3.
15:56:890Paolo Guiotto: So exactly the same questions.
15:59:620Paolo Guiotto: So let's see the solution. Number one. Again, we can say that, for example, evaluating along the x-axis, we see that we get X power 4 that goes to plus infinity. One point x00 goes to the infinity of
16:19:210Paolo Guiotto: are 3. That means norm of this vector, which is absolute value of X, goes to plus infinity.
16:26:610Paolo Guiotto: Okay, so here the right coordinates are the spherical coordinates, so the polar coordinates of space. We can see that F of XYZ in spherical
16:43:160Paolo Guiotto: coordinates.
16:44:810Paolo Guiotto: Well…
16:46:170Paolo Guiotto: for your psychological, say, safety, we write the equation, but you actually don't need to write them, because if you look at, here you have raw square.
16:57:670Paolo Guiotto: squared, so it is rho power 4. From there, you know that each of these variables is multiplied by rows. We have a rho cubed times certain number of sine and cosine. So I don't need explicitly to write which are these sine and cosine, but however, we can say that x is…
17:14:880Paolo Guiotto: So raw, what is it? Raw signed fee?
17:18:380Paolo Guiotto: Costita.
17:20:880Paolo Guiotto: Raw signed feet.
17:24:890Paolo Guiotto: sine theta.
17:27:340Paolo Guiotto: And Z is raw cost fee.
17:31:200Paolo Guiotto: So this becomes raw square to power 2, that's the X squared, Y squared plus Z square, minus XYZ is raw cube, and then we have a bunch of these products, so XY yields sine…
17:48:320Paolo Guiotto: Square phi, then cos theta, sine theta…
17:54:290Paolo Guiotto: And then we have a cost fee, again.
17:59:200Paolo Guiotto: So at the end, we see that we have, A raw fourth power.
18:05:20Paolo Guiotto: minus… raw third power, times a coefficient that depends on theta phi.
18:13:430Paolo Guiotto: Which is, since it is a product of these numbers, it is between minus 1 and 1.
18:20:870Paolo Guiotto: So…
18:22:360Paolo Guiotto: Since I want to bet, of course, on the fact that this goes to infinity, I have to say that, at worst, I will subtract exactly rho cube from rh power 4.
18:35:710Paolo Guiotto: That is sufficient to say that this goes to infinity when rho goes to plus infinity.
18:40:840Paolo Guiotto: And this says that there exists limit
18:45:500Paolo Guiotto: for F at infinity equal to plus infinity.
18:51:710Paolo Guiotto: Number two, let's look at stationary points. So, here, we can say that
18:58:520Paolo Guiotto: Well, the nice thing is that this function depends on XYZ in the same way, so once we compute the derivative, we automatically have the other derivatives. Just flip the letters.
19:08:680Paolo Guiotto: However, derivative with respect to X of F is 2, X squared plus Y squared plus Z squared. Then there is the derivative of this quantity with respect to X, which is 2X.
19:22:810Paolo Guiotto: Then I have minus XYZ, the derivative with respect to Z is YZ.
19:30:20Paolo Guiotto: So similarly, DYF will be 2 times 2, 4, there will be the letter Y here, X squared plus Y squared plus Z squared.
19:41:150Paolo Guiotto: minus X times Z, and the derivative with respect to Z will be 4Z X squared plus Y squared plus Z squared minus XY. And these are the three derivatives.
19:56:500Paolo Guiotto: Now, we can clearly see that they are all continuous functions on
20:02:230Paolo Guiotto: R3. So this says that, because of the differentiability test.
20:09:600Paolo Guiotto: that the function f is differentiable on… R3.
20:16:880Paolo Guiotto: Which is something which is underlined, but it's something that is always required to apply the theory.
20:25:280Paolo Guiotto: Particularly the Fermat theorem we are going to use later.
20:30:150Paolo Guiotto: Now, XYZ is a stationary point, is stationary point.
20:37:310Paolo Guiotto: for this F, if and all if…
20:42:510Paolo Guiotto: Gradient F is the vector 0 of our tree, so this is three equations, those three derivatives equal to 0.
20:51:940Paolo Guiotto: Okay, so 4X… you see that there is no way to factorize here anything?
20:59:550Paolo Guiotto: the three equations are the same type of equation, I don't see anything to be factorized, so I just write
21:06:630Paolo Guiotto: the equations x squared plus Y squared plus Z squared minus YZ equals 0, and the similar equation for the second and the third derivative.
21:20:650Paolo Guiotto: Now, what could we do here?
21:25:170Paolo Guiotto: Yeah, you have understood that idea, right? So we… we have this quantity, which is really disturbing, you know? It's complicated, there are squares, there are the three variables. Let's explicit, but here you have to be a little bit careful. It is indifferent which equation we choose, so let's take the first one.
21:43:510Paolo Guiotto: Now, if we take the first one, we have 4X times… Well.
21:50:820Paolo Guiotto: that quantity, X squared plus Y squared plus Z squared, equal YZ. So what I want to do is to divide by 4X. But before to do this, I can do this, of course, if X is different from 0.
22:05:170Paolo Guiotto: Okay? So for X, different from 0. So what if X is 0?
22:10:930Paolo Guiotto: Now, let's see what happens if we plug X equals 0, because this becomes another case. So I have this alternative. Either X is 0,
22:20:210Paolo Guiotto: And let's see what happens to the three equations. Or, X is different from 0, and then this identity holds, X squared plus Y squared plus Z squared equal YZ divided for X.
22:37:620Paolo Guiotto: Now, if X is 0,
22:43:310Paolo Guiotto: So what happens to the first equation? The first equation becomes, you see that de facto there kills the parentheses, so we have minus YZ equals zero.
22:55:930Paolo Guiotto: Wait, let's do things slowly, let's write, and let's see what happens.
23:00:750Paolo Guiotto: The second equation, when X is 0,
23:04:460Paolo Guiotto: simplifies a bit, we get 4Y times Y squared plus Z squared equals 0. So 4Y times Y squared plus Z square equals 0, which is, in any case, better than this one, because it's a product equals 0.
23:21:350Paolo Guiotto: And the last line is the same, because when X is 0, you see that it kills the last term, the parenthesis simplifies a bit, so we get 4Z.
23:32:10Paolo Guiotto: times Y squared plus Z squared equals 0. So this is how the system becomes in the first case.
23:40:360Paolo Guiotto: In the second case, I use that quantity, and I plug not, of course, into the first equation, because into the first equation becomes an identity 0 equals zero, you see?
23:51:460Paolo Guiotto: No? I don't need to plug this into this. No? It's useless, because it just comes from there, no?
23:59:640Paolo Guiotto: So I will get a trivial identity. I will instead plug into the second and third equation. So the second is 4Y times that block, so this will become 4Y times
24:11:470Paolo Guiotto: Now I plug this, YZ divided 4X,
24:16:450Paolo Guiotto: Minus what is de facto XZ.
24:21:730Paolo Guiotto: equals 0. And the same in the last line. So the last line is 4Z times the X squared plus Y squared plus Z squared, so you have YZ divided 4X minus XY, if I'm not wrong.
24:38:620Paolo Guiotto: Yeah.
24:40:470Paolo Guiotto: Okay, so…
24:41:780Paolo Guiotto: Seems a little bit complicated, but as you will see, we can deal. So let's now work on the left system.
24:49:350Paolo Guiotto: So here, of course, it would be… there is no big difference between these three equations, but let's work on this.
24:56:790Paolo Guiotto: This yields an alternative. Either Y is 0 or Z is 0. So we have, from this, we have X, which is already equal 0, Y equals 0, or…
25:09:300Paolo Guiotto: still x equals 0. This remains because it's part of this system.
25:14:30Paolo Guiotto: and Z equals 0.
25:16:190Paolo Guiotto: If Y is 0, and we plug into the second… in the third line there, so into this line here, you get 0 equals 0. So this is a trivial identity, which we can cancel, while when we plug Y equals 0 in the last line, we get 4Z cubed equals 0.
25:36:340Paolo Guiotto: We do the same in the second case. When Z is 0, I plug into the second line, I get 4Y cubed equals 0, while when I plug into the last line, I get 4Z, which becomes 0, equal to 0.
25:51:560Paolo Guiotto: And so this is a useless identity.
25:55:210Paolo Guiotto: Now, you see that these two are the same system, the sense that they produce the same solution, so the unique possibility here is XYZ equal .000.
26:09:420Paolo Guiotto: Okay? So this is the solution of the left side system. Let's now work on the right side.
26:16:650Paolo Guiotto: Okay, so let's perhaps write better this, so…
26:21:580Paolo Guiotto: We have a condition, which is not an equation, no, X different from 0 is not an equation, but it says this system here is working under this condition. This means that if I find at the end that X equals 0, it means that
26:39:130Paolo Guiotto: Now, if in this second case, suppose that at the end I arrive and I read X equals 0, what does it mean?
26:48:220Paolo Guiotto: that there is no solution, okay? Okay, so let's keep in mind this. Since X is 0, I can remultiply by 4X instead of
26:59:00Paolo Guiotto: in such a way that I can put in line this. So this becomes, for example, the second one becomes 4Y squareZ minus, when I multiply by 4X, becomes 4X squared Z equals 0, which is nice, because it can be factorized.
27:16:550Paolo Guiotto: And similarly, the last one is 4Y,
27:21:440Paolo Guiotto: Z square, so let's keep the same style. So Z square Y minus 4X square Y equals 0.
27:33:850Paolo Guiotto: And, do not forget that we have also that equation. X squared plus Y squared plus Z squared equal, what? X, YZ divided for X.
27:47:640Paolo Guiotto: Forgetting an equation, you forget condition, you forget constraints on solutions, you could find wrong solutions, okay? So, be very, very careful when you solve these things.
28:00:120Paolo Guiotto: So now, let's take, for example, let's divide by 4, this can be done without any problem, and let's factorize these equations. That's the second and the third, so I continue here.
28:12:940Paolo Guiotto: Okay, so my first condition is X different from 0, the second one can be, say, the Z times… you see that there is Y squared minus X squared that can be factorized, the Y minus X times Y plus X.
28:27:930Paolo Guiotto: all this equal to zero, and similarly, the next equation, which is Y times Z squared minus X squared, which is Z minus X times Z plus X.
28:40:210Paolo Guiotto: equals 0. And finally, the last line, which is X squared plus Y squared plus Z squared, equal YZ divided 4X.
28:52:700Paolo Guiotto: Okay, so now let's work on this second equation.
28:59:80Paolo Guiotto: This yields 3 sub-cases.
29:01:880Paolo Guiotto: So, I have… SubK is 1, X different from 0, and Z equals 0.
29:09:70Paolo Guiotto: Put this into the other two lines.
29:11:680Paolo Guiotto: you get in the next one, you get Y times minus X squared.
29:17:490Paolo Guiotto: Y, so minus YX squared equals 0. Since you know that X is different from 0, now, you can simplify by X squared, right?
29:28:950Paolo Guiotto: Because that number is different from 0 here, so it will be just… you don't do the alternative, I'm saying, okay? Because X squared is different from 0. And similarly, the last one, when Z is 0… no, sorry, I'm doing a mess. This is the equation. About the last one.
29:50:00Paolo Guiotto: Well, you see that here we have X different from 0, already Z equals 0 and Y equals 0. Let's use, to simplify as much as possible, the last line becomes, with both Y and Z equals 0, it becomes X squared
30:04:430Paolo Guiotto: Equal… you see that YZ is 0 now, equals 0. So what is the interpretation of this?
30:11:710Paolo Guiotto: Impossible.
30:13:620Paolo Guiotto: So this will not produce any other solution. So let's take the second case, which is X different from 0. Now we take this one, Y equal to X,
30:27:340Paolo Guiotto: Let's plug into the third line. So if Y is equal to X, I can say the third line becomes Y… sorry, X times Z minus X times Z plus X equals 0, and the last line is just a 2X squared
30:45:440Paolo Guiotto: plus Z square equal… Y is equal to X, so I can cancel. You see, Y over X is 1, so Z divided 4.
30:55:700Paolo Guiotto: So these are still to be worked out, because now, I can simplify by X, it is different from 0, so let's throw away.
31:03:980Paolo Guiotto: And then we have a third alternative, which is X different from 0. Let's see. We have still one minute, so let's try to…
31:14:890Paolo Guiotto: Y equal minus X, you see?
31:18:810Paolo Guiotto: No, no, I'm doing a… yeah, it is correct. So, it's basically the same, because this is minus X, Z minus X, Z plus X equals 0. So, again, you can't cancel this minus X, because it is different from 0.
31:36:910Paolo Guiotto: And plug in this Y equals minus X in the last line, it becomes…
31:42:830Paolo Guiotto: X squared plus Y squared is a 2X squared.
31:46:840Paolo Guiotto: plus Z square equal, now, Y is minus X, so Y divided X is minus 1, so minus Z divided 4.
31:58:450Paolo Guiotto: Well, we have Steve to work Peter.
32:01:90Paolo Guiotto: So now we work on this equation, two sub-cases, Z equal X, Z equal minus X.
32:09:640Paolo Guiotto: So, this yields.
32:14:620Paolo Guiotto: So, in other words, these type of problem are survival exercise.
32:24:110Paolo Guiotto: So we have Z equal X, and now you plug this into the last line, you get 2X squared plus another X squared, so 3X squared.
32:35:320Paolo Guiotto: Equal X divided 4.
32:38:740Paolo Guiotto: Since X is different from 0, we can simplify, so this can simplify it with this. We can divide, we are allowed to divide by X.
32:48:420Paolo Guiotto: And so, luckily, this produced something, because this produces X equal 1 over 12, right?
32:58:950Paolo Guiotto: So 1 over 12, Y is X, so another 1 over 12, and Z is X. So we have a non-trivial point.
33:08:40Paolo Guiotto: So now, it's similar to this, so here you have X different from 0, Y was already X, and Z is minus X.
33:18:960Paolo Guiotto: So when you plug this into the last line, you get, again, the 3X squared.
33:24:780Paolo Guiotto: equal Z divided 4, which is minus 1 4.
33:29:500Paolo Guiotto: So, sorry, X, I'm running a bit too much.
33:34:550Paolo Guiotto: So, this is, you simplify, you divide by 3, so 1 12, and this yields the point, minus 1 over 12,
33:43:910Paolo Guiotto: The Y is X, so minus 1 over 12, and DZ is minus X, so 1 over 12.
33:51:140Paolo Guiotto: Now, you finish by finding the solutions of the Asada case, which are similar. You will see still this 1 over 12 with some plus-minus sign, okay?
34:01:800Paolo Guiotto: Okay, let's stop here, and tomorrow we completed the exercise.
34:09:890Paolo Guiotto: Very exciting.