0:00:01.230,0:00:06.742 In this video, we're going to be[br]looking at how curves behave and 0:00:06.742,0:00:09.286 how to find specific points on 0:00:09.286,0:00:16.550 those curves. Let's begin by[br]looking at a sketch of a curve. 0:00:21.230,0:00:26.906 Now we've got a curve. There's[br]some noticeable things about it. 0:00:27.460,0:00:29.048 First of all here. 0:00:30.860,0:00:38.290 The curve turns if we[br]draw tangent to the curve. 0:00:38.840,0:00:44.972 It's level parallel to the X[br]axis. Here I've exaggerated it a 0:00:44.972,0:00:50.593 little bit. The curve is flat[br]and the tangent is again 0:00:50.593,0:00:53.148 parallel to the X axis. 0:00:53.720,0:01:01.700 Here the curve turns and[br]again a tangent is parallel 0:01:01.700,0:01:04.892 to the X axis. 0:01:05.780,0:01:12.150 Now, one of the things that we[br]do know is that the gradient of 0:01:12.150,0:01:18.975 a curve is given by DY by The[br]X. What's clear is that each of 0:01:18.975,0:01:20.340 these points here. 0:01:21.310,0:01:23.428 Here and here. 0:01:24.030,0:01:29.761 Because the tangent is flat[br]parallel to the X axis, the 0:01:29.761,0:01:35.492 gradient is zero and so D why[br]by DX is 0. 0:01:36.360,0:01:43.080 Now for points like these where[br]DY by the X is zero, we have 0:01:43.080,0:01:45.960 a special name, we call them. 0:01:46.920,0:01:50.015 Stationary 0:01:50.015,0:01:56.407 points. If[br]you like the curve, sort of 0:01:56.407,0:02:02.059 isn't moving, it isn't changing.[br]At that point. The why by DX, 0:02:02.059,0:02:05.356 which is the rate of change, is 0:02:05.356,0:02:11.920 0. Now two of these I discribed[br]the curve as turning. I said 0:02:11.920,0:02:17.080 there the curb was turning and[br]here again the curve is turning 0:02:17.080,0:02:22.670 so that's going from coming up[br]to going down and this one is 0:02:22.670,0:02:28.260 from going down to coming up.[br]This one isn't this one is very 0:02:28.260,0:02:29.980 different so these two. 0:02:30.910,0:02:33.740 Are special types of stationary 0:02:33.740,0:02:36.670 points. These two. 0:02:37.450,0:02:41.056 Are cold turning 0:02:41.056,0:02:47.530 points. So at[br]turning points divided 0:02:47.530,0:02:50.692 by DX is 0:02:50.692,0:02:57.932 0. Thing to realize though, is[br]that if the wide by the X is 0:02:57.932,0:03:02.486 zero, that doesn't mean we've[br]got a turning point. We might 0:03:02.486,0:03:07.454 have one of these, that's what[br]makes life just a little bit 0:03:07.454,0:03:11.180 tricky, but nevertheless thing[br]to remember is that turning 0:03:11.180,0:03:16.562 points divided by DX is 0. Now[br]let's have a look at the 0:03:16.562,0:03:19.046 nature of this particular[br]turning point. 0:03:20.590,0:03:23.836 In the immediate area of it. 0:03:24.980,0:03:30.846 The curve has no bigger value.[br]We go up, we reach a value and 0:03:30.846,0:03:33.779 then we go down again. So this 0:03:33.779,0:03:40.400 point. Is called[br]a local maximum. 0:03:40.960,0:03:43.688 Because in the immediate 0:03:43.688,0:03:46.688 vicinity. There is no bigger 0:03:46.688,0:03:53.178 value. Now usually we forget[br]about the local and we just call 0:03:53.178,0:03:57.958 it a maximum. Similarly, this[br]point is called a local. 0:04:00.740,0:04:06.695 Minimum. Because again, in this[br]particular region of it, the 0:04:06.695,0:04:13.905 curve goes no lower, so this is[br]the least value in this area are 0:04:13.905,0:04:19.934 local minimum. Again, we forget[br]the word local quite often. 0:04:19.934,0:04:26.210 Unjust use minimum. In essence,[br]this video is going to be about 0:04:26.210,0:04:27.779 finding these points. 0:04:28.400,0:04:34.640 Maxima and minima finding these[br]two points Maxima and minima. 0:04:35.180,0:04:38.284 And distinguishing between[br]them because being able to 0:04:38.284,0:04:43.328 find them isn't a great deal[br]of use unless we can tell them 0:04:43.328,0:04:47.596 apart. So we need to be able[br]to distinguish between a 0:04:47.596,0:04:51.476 maximum and the minimum. So[br]let's begin by exploring what 0:04:51.476,0:04:53.416 happens around a minimum[br]point. 0:04:54.670,0:04:56.440 Draw a minimum. 0:04:57.410,0:05:00.618 Let's take the Tangent. 0:05:02.090,0:05:07.161 At the minimum point, we know[br]that the gradient of this 0:05:07.161,0:05:13.154 tangent is 0 because it's[br]parallel to the X axis. So DY by 0:05:13.154,0:05:14.537 DX is 0. 0:05:15.170,0:05:19.618 Let's take a point a[br]little bit before. 0:05:21.670,0:05:26.470 The minimum point and take a[br]tangent at that point. And here 0:05:26.470,0:05:28.470 we see that the Tangent. 0:05:30.250,0:05:36.886 Is negative, it's got a negative[br]slope and so D. Why by DX is 0:05:36.886,0:05:41.626 negative. I'm using this as a[br]shorthand for negative, not 0:05:41.626,0:05:47.314 actually bothered about the[br]value of the why by DX just it's 0:05:47.314,0:05:52.054 sign. Let's take a point here a[br]little bit after. 0:05:52.890,0:05:57.858 What's happened there? Well, we[br]can see we've got a line that's 0:05:57.858,0:06:03.240 got a positive slope to it,[br]which means that the why by the 0:06:03.240,0:06:05.724 X has got to be positive. 0:06:06.700,0:06:13.190 So what's happening as we pass[br]through this point as X 0:06:13.190,0:06:17.910 increases? What's happening to[br]do, why by DX? 0:06:18.760,0:06:24.326 Well, as X increases, the why by[br]DX goes from negative. 0:06:25.480,0:06:28.460 0. To positive. 0:06:29.620,0:06:36.473 So divide by the ex is[br]increasing as X increases, so 0:06:36.473,0:06:42.703 let's write that down deep. Why[br]by DX is increasing. 0:06:46.270,0:06:48.820 As X. 0:06:51.300,0:06:58.308 Increases. What[br]does that mean? DY by DX is 0:06:58.308,0:07:02.637 increasing well that means that[br]the rate of change. 0:07:03.150,0:07:05.660 Of the why by DX? 0:07:06.880,0:07:11.910 Is[br]positive. 0:07:13.670,0:07:21.302 IE D2Y by[br]DX squared is 0:07:21.302,0:07:25.118 greater than 0. 0:07:27.470,0:07:32.750 So if D2Y by the X squared is[br]greater than 0. 0:07:33.550,0:07:39.892 And divide by the X is equal[br]to 0. We know that we have 0:07:39.892,0:07:41.704 got a minimum point. 0:07:43.130,0:07:47.267 Is that true for all[br]minimum points? 0:07:48.500,0:07:54.044 Little trick of logic here, but[br]just write down what we've got 0:07:54.044,0:07:57.706 if. DY by DX 0:07:57.706,0:08:01.000 equals 0. And. 0:08:02.390,0:08:08.011 D2Y by DX squared[br]is greater than 0:08:08.011,0:08:10.420 or positive then. 0:08:11.820,0:08:14.880 We have 0:08:14.880,0:08:19.520 a.[br]Minimum. 0:08:20.650,0:08:23.974 Does it work the other way[br]around? If we have a minimum? 0:08:24.950,0:08:29.474 Do these two both apply? Well,[br]certainly that one does that if 0:08:29.474,0:08:34.752 we have a minimum DY by the X[br]will be 0, but not necessarily 0:08:34.752,0:08:41.842 this one. Let's have a look[br]at the curve Y equals X to 0:08:41.842,0:08:45.610 the 4th. If I sketch this curve. 0:08:47.030,0:08:47.920 It looks. 0:08:51.010,0:08:51.800 Like that? 0:08:53.830,0:08:58.906 Here we've clearly got our[br]turning point. Our minimum 0:08:58.906,0:09:01.775 point. Let's have a look what's 0:09:01.775,0:09:09.349 happening. DY by the X equals[br]well. Why was X to the 4th so 0:09:09.349,0:09:15.758 DY by DX is 4X cubed, and we[br]know that for our stationary 0:09:15.758,0:09:22.660 points we put it equal to 0,[br]which tells us X equals 0 and 0:09:22.660,0:09:29.562 X equals 0 tells us that Y[br]equals 0, so we know that this 0:09:29.562,0:09:31.041 is a stationary. 0:09:32.130,0:09:33.110 Point 0:09:34.190,0:09:39.536 What kind of stationary point is[br]it? Let's do our second 0:09:39.536,0:09:46.780 derivative. D2 why by DX[br]squared is 12 X squared, but 0:09:46.780,0:09:48.604 it equals 0. 0:09:49.170,0:09:52.478 When X equals 0. 0:09:54.190,0:09:56.788 So this doesn't work in reverse. 0:09:57.480,0:10:02.744 If the why by DX is zero and Y2Y[br]by the X squared is greater than 0:10:02.744,0:10:05.047 not, then we know we have a 0:10:05.047,0:10:09.620 minimum. But it doesn't work[br]going the other way. What that 0:10:09.620,0:10:13.635 means is that we have to be[br]especially careful whenever D2, 0:10:13.635,0:10:18.745 why? By DX squared comes out to[br]be 0 when we're doing the test, 0:10:18.745,0:10:19.840 so to speak. 0:10:20.610,0:10:24.419 We need to look at it in a[br]slightly different way. In fact, 0:10:24.419,0:10:28.814 what we need to do is go back to[br]basics and I'll show you an 0:10:28.814,0:10:32.330 example later where we do go[br]back to basics and how that 0:10:32.330,0:10:38.122 works out. So having had a look[br]at a minimum point and looked at 0:10:38.122,0:10:42.815 it in some detail, let's now[br]move on and look at a maximum 0:10:42.815,0:10:44.620 point and see what happens 0:10:44.620,0:10:48.480 there. So here's[br]our maximum point. 0:10:49.940,0:10:55.010 And at the point itself, we know[br]that the tangent is parallel to 0:10:55.010,0:10:57.740 the X axis and so its gradient. 0:10:58.280,0:11:01.296 And hence the why by DX is 0. 0:11:02.350,0:11:06.430 Let's take a point a little bit[br]before that there. 0:11:07.670,0:11:12.734 We can see that we've got a[br]line which has a positive 0:11:12.734,0:11:17.798 slope to it, and so DY by the[br]X is positive there. 0:11:19.330,0:11:23.826 Now let's take somewhere just[br]past the point. 0:11:24.340,0:11:27.148 And if we have a look at[br]this week and see we've 0:11:27.148,0:11:27.850 got a tangent. 0:11:29.860,0:11:36.464 Where the slope is negative and[br]so divide by the X is negative. 0:11:37.630,0:11:43.114 So what's happening as we pass[br]through this maximum point as X 0:11:43.114,0:11:46.770 is increasing? What's happening[br]today, why by DX? 0:11:47.990,0:11:55.107 Well, divide by the axe goes[br]from being positive to being 0:11:55.107,0:12:02.224 zero to being negative. So as[br]X increases, divided by DX 0:12:02.224,0:12:06.198 decreases. As X increases. 0:12:07.480,0:12:11.854 Why[br]by 0:12:11.854,0:12:14.041 DX 0:12:14.041,0:12:21.420 decreases? Does that mean[br]it must mean that the rate of 0:12:21.420,0:12:24.534 change of the why by DX? 0:12:25.080,0:12:28.500 Is. 0:12:30.130,0:12:36.022 Negative. As X increases as we[br]go through this point at this 0:12:36.022,0:12:41.931 point. An the derivative[br]divide by the X is negative. 0:12:41.931,0:12:47.521 In other words, D2Y by the X[br]squared is negative. 0:12:49.230,0:12:53.523 So let's just write down again[br]what we've got. 0:12:55.000,0:13:01.150 If. Divide by[br]DX equals 0. 0:13:02.150,0:13:07.680 And. D2 why[br]by DX squared. 0:13:08.330,0:13:11.490 Is less than 0. 0:13:13.820,0:13:20.828 Then we have[br]a maximum point. 0:13:20.830,0:13:24.724 Again, just the same argument[br]applies. Does it work back the 0:13:24.724,0:13:29.680 same way? Answer no, it doesn't.[br]And the problem is what do we do 0:13:29.680,0:13:33.928 about D2? Why? By DX squared[br]being zero? Well, as I said, 0:13:33.928,0:13:38.884 will show a way of dealing with[br]that in a moment. It's time now 0:13:38.884,0:13:43.840 to actually look at an example[br]or two and see if we can make 0:13:43.840,0:13:45.256 use of this test. 0:13:45.810,0:13:52.818 In order to sort out the[br]turning points on a curve. So 0:13:52.818,0:13:59.826 the first example we're going to[br]have a look at is Y 0:13:59.826,0:14:03.330 equals X cubed, minus three X 0:14:03.330,0:14:09.417 +2. The first thing we[br]need to do is differentiated. 0:14:09.417,0:14:15.060 Why by DX equals if we[br]differentiate this we have three 0:14:15.060,0:14:21.216 X squared. We multiply by the[br]index and subtract 1 prom it 0:14:21.216,0:14:26.859 minus the derivative of the[br]three axis, just three, and this 0:14:26.859,0:14:29.937 is equal to 0 for stationary. 0:14:31.320,0:14:37.366 Points. We can[br]factorize this. There's a common 0:14:37.366,0:14:43.840 factor of three leaves us with X[br]squared minus one, and this is 0:14:43.840,0:14:48.820 now recognizable as the[br]difference of two squares, so it 0:14:48.820,0:14:53.302 factorizes us three times, X[br]minus 1X plus one. 0:14:53.330,0:14:54.840 And that's equal to 0. 0:14:55.430,0:14:57.488 Now what we can see is that. 0:14:58.020,0:15:04.642 Either this bracket is 0 or this[br]bracket is 0, or they're both 0, 0:15:04.642,0:15:09.845 so we have X equals 1 or X[br]equals minus one. 0:15:10.910,0:15:16.084 Now this is just given us the X[br]coordinates of the points that 0:15:16.084,0:15:21.258 were interested in what we need[br]now or the Y coordinates, and so 0:15:21.258,0:15:26.034 let's work those out, X equals[br]1, Y equals and will substitute 0:15:26.034,0:15:32.402 it in on the very first line. So[br]that's one cubed is just 1 - 3 0:15:32.402,0:15:35.586 times by one is minus 3 + 2. 0:15:35.640,0:15:38.370 Altogether, that gives us 0. 0:15:39.150,0:15:46.506 And so our point, our stationary[br]point is 1 zero. If we 0:15:46.506,0:15:52.636 take X equals minus one and[br]Y is equal to. 0:15:53.200,0:15:59.427 And we put minus one in there[br]well minus one cubed is just 0:15:59.427,0:16:06.612 minus 1 - 3 times by minus one[br]is plus three and plus two at 0:16:06.612,0:16:14.276 the end, and altogether 3 + 2 is[br]5 takeaway, one is 4 and so our 0:16:14.276,0:16:16.671 other stationary point is minus 0:16:16.671,0:16:19.980 1 four. Now we need to decide. 0:16:20.640,0:16:24.924 All these points, Maxima, minima[br]or that funny one where it was 0:16:24.924,0:16:30.279 flat. So in order to sort that[br]out, what we need to have a look 0:16:30.279,0:16:32.064 at is the 2nd derivative. 0:16:32.950,0:16:37.262 So I'm going to write the[br]equation of the curve down 0:16:37.262,0:16:41.182 again, and then I'm going to[br]write down this first 0:16:41.182,0:16:44.710 derivative, then the points, and[br]then the second derivative. 0:16:45.800,0:16:53.490 So we've Y equals X[br]cubed, minus three X +2. 0:16:53.490,0:17:01.180 We've got the first derivative[br]D. Why by DX3 X 0:17:01.180,0:17:08.870 squared minus three and the[br]points that were interested in 0:17:08.870,0:17:15.791 with the points one not[br]and minus 1 four. 0:17:15.830,0:17:21.989 So now. Let's take the second[br]derivative D2Y by DX squared 0:17:21.989,0:17:26.408 equals. So we differentiate[br]this. The derivative of the 0:17:26.408,0:17:32.300 first term we multiply by the[br]index. So two, freezer 6 and 0:17:32.300,0:17:38.683 take one of the index that's X[br]and the derivative of three is 0:17:38.683,0:17:44.712 just zero. So let's now look at[br]the points and. It's the values 0:17:44.712,0:17:47.456 of X we're interested in. So X 0:17:47.456,0:17:54.990 equals 1. D2Y by[br]DX squared equals 6. 0:17:56.360,0:17:59.390 So this is 0:17:59.390,0:18:06.690 positive. What do we[br]know? We know that if divided by 0:18:06.690,0:18:13.762 DX is zero and if D2, why[br]by DX squared is positive, then 0:18:13.762,0:18:19.746 the point we're looking at is a[br]minimum turning point, so 0:18:19.746,0:18:23.010 therefore 10 is a minimum mean 0:18:23.010,0:18:30.120 for short. X equals minus[br]one. This is our second point. 0:18:30.620,0:18:36.100 D2Y by the X squared is equal[br]to minus 6. 0:18:37.930,0:18:41.998 And that is negative. The[br]value of these doesn't 0:18:41.998,0:18:45.614 matter. It's their sign[br]that matters and therefore, 0:18:45.614,0:18:51.038 what do we know? We know[br]that if the Wi-Fi DX is 0:18:51.038,0:18:56.010 zero and Y2Y by DX squared[br]is negative, then we have 0:18:56.010,0:19:00.530 got a maximum, so minus 1[br]four is a Max. 0:19:01.660,0:19:06.820 One of the things that this does[br]help us do is when we do know 0:19:06.820,0:19:11.292 the turning points. We can use[br]this to help us gain a picture 0:19:11.292,0:19:15.764 of the graph to help us sketch a[br]picture. So let's just plot 0:19:15.764,0:19:16.796 these two points. 0:19:19.120,0:19:24.230 And see if by plotting them we[br]can build up a picture of the 0:19:24.230,0:19:26.420 graph. Now this won't be an 0:19:26.420,0:19:30.900 exact picture. The scales will[br]be different. They'll be a 0:19:30.900,0:19:34.410 little bit cramped in some[br]cases, expanded in others 0:19:34.410,0:19:38.700 because what we're trying to[br]show the important points on the 0:19:38.700,0:19:42.990 graph, so we've one and minus[br]one. Where are two important 0:19:42.990,0:19:45.720 values of X, and we know that 0:19:45.720,0:19:52.128 one 0. Was one of our turning[br]points and if we just go back, 0:19:52.128,0:19:57.216 we know that one zero was a[br]minimum turning point, so we 0:19:57.216,0:20:01.456 know that at this point the[br]curve looks like that. 0:20:02.410,0:20:08.034 The other point was minus[br]1 four there. 0:20:09.030,0:20:15.862 And we know that at that point[br]the curve was like that. It was 0:20:15.862,0:20:22.230 a maximum. The other thing we[br]know is that there are no 0:20:22.230,0:20:27.092 turning points anywhere else.[br]These are the only ones, and so 0:20:27.092,0:20:33.280 effectively we can join up the[br]curve and so that we get a rough 0:20:33.280,0:20:38.142 picture that it looks something[br]like that. We can find this 0:20:38.142,0:20:43.004 point of course by substituting[br]X equals 0. Remember, our curve 0:20:43.004,0:20:47.866 was Y equals X cubed minus 3X,[br]plus 2X is 0. 0:20:47.900,0:20:50.318 Then this must be 2 here. 0:20:50.840,0:20:55.388 This we can find bike waiting[br]why to zero and solving but 0:20:55.388,0:20:59.936 we're not interested in that at[br]the moment. It's the idea of 0:20:59.936,0:21:04.105 using this maximum and minimum[br]turning points in order to help 0:21:04.105,0:21:07.895 us gain a picture of the shape[br]of the curve. 0:21:08.990,0:21:15.374 We're going to have a look at an[br]example now where the method we 0:21:15.374,0:21:21.758 use is one that enables us to go[br]back to 1st principles, and this 0:21:21.758,0:21:28.142 is also the method that we would[br]use if D2. Why? By DX squared 0:21:28.142,0:21:30.422 turned out to be 0. 0:21:30.960,0:21:34.040 So it's a very useful method to[br]have at your fingertips. 0:21:34.630,0:21:35.938 So let's begin. 0:21:36.720,0:21:43.683 We're going to be looking at[br]finding the turning points of 0:21:43.683,0:21:45.582 this particular curve. 0:21:45.590,0:21:49.040 X minus one all squared over 0:21:49.040,0:21:56.792 X. Well it's a U over[br]a V. It's a quotient, so we need 0:21:56.792,0:22:02.600 to be able to differentiate as[br]though it were a quotient. So 0:22:02.600,0:22:08.408 let's remember how we do that.[br]We take the bottom the X. 0:22:09.020,0:22:14.660 And we multiply X by the[br]derivative of U what's on top? 0:22:14.660,0:22:17.950 So that's two times X minus one. 0:22:18.500,0:22:23.135 Times the derivative of[br]what's inside a one and 0:22:23.135,0:22:29.315 then minus. And we take UX[br]minus one 4 squared and we 0:22:29.315,0:22:33.950 multiply it by the[br]derivative of E. What's on 0:22:33.950,0:22:38.585 the bottom? And that's the[br]derivative of X. That's 0:22:38.585,0:22:43.220 just one, and then it's all[br]over X squared. 0:22:44.390,0:22:50.132 And for our stationary points we[br]put this equal to 0. 0:22:51.560,0:22:52.230 Now. 0:22:53.920,0:22:58.795 Let's have a look at this. We[br]need to tidy it up. It looks a 0:22:58.795,0:23:03.020 bit of a mess, so let's have a[br]look for some common factors. 0:23:03.020,0:23:07.570 Well, here we've got an X minus[br]one and here as well. So let's 0:23:07.570,0:23:12.120 take that X minus one out as a[br]factor. Let's have a look what 0:23:12.120,0:23:16.345 we're left with. Well, here[br]we've got two and X giving us 2X 0:23:16.345,0:23:18.295 and one times by one. That's 0:23:18.295,0:23:23.816 still 2X. Minus and here we've[br]got one times by now we just 0:23:23.816,0:23:28.834 took out X minus one as a[br]factor, so it's minus X minus 0:23:28.834,0:23:33.466 one and notice I've kept it in[br]the bracket because it's the 0:23:33.466,0:23:36.168 whole of X minus one that I'm 0:23:36.168,0:23:43.350 taking away. Close that bracket[br]all over X squared so this 0:23:43.350,0:23:46.600 is still DY by DX. 0:23:47.530,0:23:54.535 Now I can simplify what's in the[br]bracket and have DY by the X is 0:23:54.535,0:24:01.073 equal to X minus one times by[br]now have two X takeaway X that's 0:24:01.073,0:24:08.078 just an X, but then I've also to[br]take away minus one and so that 0:24:08.078,0:24:09.946 gives me plus one. 0:24:10.010,0:24:15.758 All over X squared and[br]this is to be equal to 0 0:24:15.758,0:24:17.674 for my stationary points. 0:24:18.730,0:24:23.662 Now when we have an expression[br]like that that's equal to 0, 0:24:23.662,0:24:27.772 it's only the numerator that[br]we're interested in. It's only 0:24:27.772,0:24:31.060 the numerator that will make[br]this expression 0. 0:24:31.700,0:24:37.844 So got is[br]X minus 1X 0:24:37.844,0:24:40.916 plus one equals 0:24:40.916,0:24:47.030 0. And that tells us that[br]either this bracket is equal 0:24:47.030,0:24:53.204 to 0 or that bracket is equal[br]to 0. In other words, X minus 0:24:53.204,0:24:59.378 one equals 0 or X Plus one[br]equals 0. So X equals 1. Four 0:24:59.378,0:25:01.142 X equals minus one. 0:25:02.310,0:25:08.327 Now it's the points where after[br]so let's now calculate the 0:25:08.327,0:25:15.438 values of Y, remembering that Y[br]is equal to X minus one 4 0:25:15.438,0:25:17.626 squared all over X. 0:25:18.690,0:25:24.710 So let's take X equals 1 and see[br]what that gives us for why? 0:25:25.420,0:25:31.885 We put one in there. We have 1 -[br]1 all squared over one. Well, 0:25:31.885,0:25:38.781 that's just zero because 1 - 1[br]is 0 square at that still 0 / 1. 0:25:38.781,0:25:44.815 The answer is still 0. So one of[br]our points is one Nord. That's 0:25:44.815,0:25:49.987 one of our stationary points.[br]Now let's do X equals minus one. 0:25:50.590,0:25:58.446 Why is equal to minus 1 - 1[br]all squared all over minus 1 - 1 0:25:58.446,0:26:05.811 - 1 is minus 2 and we square[br]it to that's four but we divide 0:26:05.811,0:26:12.685 by minus one, so that is minus[br]four. So other point is minus 1 0:26:12.685,0:26:18.194 - 4. So we've got two[br]points which are stationary 0:26:18.194,0:26:24.408 points and we need to find out[br]are they turning points and if 0:26:24.408,0:26:28.710 they are turning points, are[br]they Maxima or minima? 0:26:29.560,0:26:34.636 So let's write down[br]again. Why do you? Why 0:26:34.636,0:26:37.456 by DX and our points. 0:26:38.680,0:26:45.780 So we had why that[br]was X minus one or 0:26:45.780,0:26:47.910 squared over X. 0:26:48.500,0:26:55.611 We had DY by the X&YYY DX[br]was over X squared and let's 0:26:55.611,0:26:59.440 just go back so we can see 0:26:59.440,0:27:06.660 what. The numerator was it was X[br]minus one times by X plus one. 0:27:07.370,0:27:14.240 X minus one times by[br]X Plus One and our 0:27:14.240,0:27:21.250 two points. Are the[br]points one note and minus 0:27:21.250,0:27:23.440 1 - 4? 0:27:24.410,0:27:25.590 Now. 0:27:26.640,0:27:31.619 In order to sort out if these[br]are turning points and if they 0:27:31.619,0:27:36.598 are, which is a maximum and[br]which is a minimum, then we need 0:27:36.598,0:27:40.428 to differentiate this again and[br]that is very very fearsome. 0:27:41.420,0:27:42.920 Very fearsome indeed. 0:27:43.790,0:27:45.980 Is it really worth[br]differentiating 0:27:45.980,0:27:49.046 differentiating it[br]again? We might make a 0:27:49.046,0:27:51.674 mistake. The algebra[br]might go wrong. 0:27:52.740,0:27:57.833 So rather than differentiating[br]it again, let's do what we did 0:27:57.833,0:28:02.926 before. Let's look at the[br]gradient a little bit before X 0:28:02.926,0:28:09.871 equals 1. At X equals 1 and a[br]little bit after X equals 1 and 0:28:09.871,0:28:15.890 form a picture from doing that[br]of whether this is a maximum or 0:28:15.890,0:28:21.446 minimum point, and this method[br]that we're going to have a look 0:28:21.446,0:28:22.835 at also works. 0:28:22.890,0:28:27.776 Wendy to why? By DX squared is[br]0, so hence it's a good method 0:28:27.776,0:28:31.964 and it's well worth adding it to[br]your Armory of techniques for 0:28:31.964,0:28:36.152 looking at stationary points. So[br]let's set this up and the way 0:28:36.152,0:28:38.944 we're going to set it up is too. 0:28:39.890,0:28:43.238 Have a little table. 0:28:43.240,0:28:47.025 So here's the Y by 0:28:47.025,0:28:53.580 The X. And here is[br]the point where interested in at 0:28:53.580,0:28:54.990 X equals 1. 0:28:56.150,0:29:02.562 I want to look a little bit[br]before X equals 1, so I'm going 0:29:02.562,0:29:08.974 to say this is X equals 1 minus[br]Epsilon and all. I mean by 0:29:08.974,0:29:11.722 Ipsilon is a little positive bit 0:29:11.722,0:29:17.007 of X. North Point One North[br]Point North one if you like, and 0:29:17.007,0:29:19.226 I'm also going to look at the 0:29:19.226,0:29:25.714 gradient. A little bit after X[br]equals 1, so let me call that 0:29:25.714,0:29:27.115 one plus Epsilon. 0:29:28.470,0:29:35.036 Now I'm not interested in the[br]value of the why by DX. I'm only 0:29:35.036,0:29:39.726 interested in its sign whether[br]it's positive or negative. I 0:29:39.726,0:29:44.885 know that when X equals 1[br]divided by DX is 0. 0:29:46.290,0:29:53.970 What happens if I am a[br]little bit before X equals 1? 0:29:55.200,0:30:00.645 X is a little bit less than one,[br]so I have something a little bit 0:30:00.645,0:30:02.823 less than one, and I'm taking 0:30:02.823,0:30:08.710 one away. So the answer to that[br]is negative. This bracket here 0:30:08.710,0:30:13.178 is negative. However, I've got[br]something that's a little bit 0:30:13.178,0:30:15.374 less than one plus one that's 0:30:15.374,0:30:19.756 positive. So I have a[br]negative times by a 0:30:19.756,0:30:23.976 positive and I've got X[br]squared on the bottom here 0:30:23.976,0:30:28.196 and X squared is positive,[br]so I'm negative times a 0:30:28.196,0:30:31.572 positive divided by a[br]positive is a negative. 0:30:33.030,0:30:38.084 Let's have a look when X is a[br]little bit more than one little 0:30:38.084,0:30:42.416 bit bigger than one 1.911. Take[br]away one. Well, a little bit 0:30:42.416,0:30:47.109 more than one. We set. Take away[br]one that's positive. One plus a 0:30:47.109,0:30:49.275 little bit more than one that's. 0:30:49.890,0:30:53.256 Positive again, and this is[br]definitely positive in the 0:30:53.256,0:30:57.744 denominator, so we've a positive[br]times by a positive divided by a 0:30:57.744,0:30:59.614 positive gives us a positive. 0:31:00.180,0:31:01.928 So if we sketch. 0:31:02.540,0:31:08.656 The slopes of the tangents. We[br]have a negative slope aflat 0:31:08.656,0:31:15.328 slope under positive slope and[br]So what we can see is that 0:31:15.328,0:31:22.000 this defines the shape of a[br]minimum and so we can conclude 0:31:22.000,0:31:25.336 that one zero is a minimum. 0:31:25.370,0:31:31.337 Point notice what we've done is[br]go back to basics. This is how 0:31:31.337,0:31:36.386 we arrived at our test for D2.[br]Why? By DX squared. 0:31:37.340,0:31:43.376 Now let's do it again for this[br]point, minus 1 - 4. 0:31:44.850,0:31:52.134 So we've got DYIDX is equal[br]to X minus one times by 0:31:52.134,0:31:58.811 X Plus one all over X[br]squared, and the point we're 0:31:58.811,0:32:06.095 having a look at is minus[br]1 - 4. So let's set 0:32:06.095,0:32:08.523 up our table again. 0:32:09.960,0:32:14.960 We're looking at the sign of DY[br]by The X. 0:32:15.490,0:32:21.301 My looking at it at the point[br]X equals minus one. As we 0:32:21.301,0:32:26.665 pass through that point as X[br]increases. So we want to know 0:32:26.665,0:32:30.688 what's happening alittle bit[br]before that minus one minus 0:32:30.688,0:32:31.135 Epsilon. 0:32:32.180,0:32:38.076 And we want to know what's[br]happening alittle bit after that 0:32:38.076,0:32:40.220 minus one plus Epsilon. 0:32:41.070,0:32:47.625 Well, we know that when X is[br]minus one DY by DX is 0. Because 0:32:47.625,0:32:49.810 this factor here is 0. 0:32:51.440,0:32:57.356 What happens if X is a little[br]bit less than minus one? 0:32:57.356,0:32:59.328 Something like minus 1.1? 0:33:00.080,0:33:05.000 Well you got minus 1.1 takeaway.[br]One that's definitely negative. 0:33:06.200,0:33:11.840 Minus 1.1 at on. One that's[br]minus .1. That's definitely 0:33:11.840,0:33:13.532 negative as well. 0:33:14.300,0:33:18.810 However, on the bottom here,[br]we've got a number and we're 0:33:18.810,0:33:23.320 squaring it, so it must be[br]positive. So we've a negative 0:33:23.320,0:33:27.830 times a negative, which is a[br]positive divided by a positive. 0:33:27.830,0:33:30.290 So the answer must be positive. 0:33:31.260,0:33:36.945 Now let's have a look a little[br]bit after X equals minus 1 - 1, 0:33:36.945,0:33:38.461 plus a little bit. 0:33:39.460,0:33:45.438 So the little bit might be not[br].1 and minus one plus a little 0:33:45.438,0:33:52.270 bit would then be minus N .9. So[br]we've got minus 9.9 - 1 - 1.9 0:33:52.270,0:33:57.730 it's negative. I've got minus[br]not .9 plus one. Well, that's 0:33:57.730,0:34:03.130 not .1, it's positive, so have a[br]negative times a positive and 0:34:03.130,0:34:08.080 this term X squared is always[br]positive 'cause it's a square. 0:34:08.080,0:34:13.930 So we've got a negative times by[br]a positive and divided by a 0:34:13.930,0:34:16.180 positive. So I'll answer. Must 0:34:16.180,0:34:23.044 be negative. So let's sketch the[br]shape. We've got a positive 0:34:23.044,0:34:29.004 slope, aflat slope and negative[br]slope, and this defines the 0:34:29.004,0:34:36.156 shape of a maximum, and so[br]we can say minus 1 - 0:34:36.156,0:34:38.540 4 is a maximum. 0:34:38.540,0:34:45.790 Finally, let's just see how[br]this helps us to sketch 0:34:45.790,0:34:52.715 the curve. We recall[br]the equation of the curve 0:34:52.715,0:34:59.765 Y equals X minus one[br]all squared over X, and 0:34:59.765,0:35:06.110 we had a maximum point[br]at minus 1 four. 0:35:07.020,0:35:12.096 And we had a minimum[br]point at one 0. 0:35:14.100,0:35:20.946 So some axes, let's mark the .1[br]and the point minus one goes on 0:35:20.946,0:35:28.281 the X axis. Now 10 is here and[br]we know it's a minimum, so we 0:35:28.281,0:35:34.638 know it looks something like[br]that there minus 1 - 4 down here 0:35:34.638,0:35:40.506 somewhere. And we know that[br]that's a maximum, so it kind of 0:35:40.506,0:35:41.973 looks like that. 0:35:43.430,0:35:47.533 How can we join these up? Well,[br]clearly there's something odd 0:35:47.533,0:35:52.382 going on around 0 here because[br]the curve seems to go that way 0:35:52.382,0:35:56.485 and that way it seems to go in[br]opposite directions. What's 0:35:56.485,0:36:00.588 happening here? What's going on?[br]Well, let's just have a little 0:36:00.588,0:36:03.945 think about this as X approaches[br]0 from above. 0:36:04.620,0:36:07.896 And this is always going to be[br]positive, but this is going to 0:36:07.896,0:36:12.380 be. Positive as well, and were[br]divided by something very very 0:36:12.380,0:36:17.138 small. Our answer is going to be[br]very, very big, so we're going 0:36:17.138,0:36:21.896 to have something up there when[br]X is negative but near to 0. 0:36:21.896,0:36:25.556 This is still going to be[br]positive because we're squaring 0:36:25.556,0:36:29.582 it, but we're dividing it by[br]something which is getting very 0:36:29.582,0:36:34.340 very small, but negative. So the[br]answer is going to be very big, 0:36:34.340,0:36:39.098 but negative, and so it will be[br]somewhere down here. So what we 0:36:39.098,0:36:42.673 can see. Is that the curve is[br]going to look something? 0:36:43.240,0:36:44.200 Like 0:36:45.930,0:36:50.324 That and they will in fact be[br]this almost what looks like a 0:36:50.324,0:36:55.056 sort of hole in the middle of[br]it. Bang on this value X equals 0:36:55.056,0:37:02.914 0. So what have we done?[br]We have found out how to find 0:37:02.914,0:37:07.109 stationary points. Stationary[br]points occur on a curve. Wendy 0:37:07.109,0:37:08.944 why by DX is 0. 0:37:09.990,0:37:13.458 If the why by DX is 0:37:13.458,0:37:19.495 0? Then some of those points are[br]what we call turning points, 0:37:19.495,0:37:20.830 Maxima or minima. 0:37:21.870,0:37:23.650 We can find those. 0:37:24.170,0:37:31.123 In one of two ways, if the why[br]by DX is zero and D2, why by DX 0:37:31.123,0:37:36.031 squared is greater than not,[br]then we know for sure that we 0:37:36.031,0:37:38.076 have got a minimum point. 0:37:39.130,0:37:45.310 If you divide by the X is zero[br]and D2 why by DX squared is 0:37:45.310,0:37:50.254 negative, then we know for sure[br]we've got a maximum point. If 0:37:50.254,0:37:55.610 however, D2 why by DX squared is[br]0, then we have no information 0:37:55.610,0:38:00.142 whatsoever and we need to look[br]very closely using the methods 0:38:00.142,0:38:02.202 that I've just shown here. 0:38:03.470,0:38:07.122 In order to determine what kind[br]of stationary point we've got, 0:38:07.122,0:38:10.774 whether we've got a maximum,[br]whether we've got a minimum or 0:38:10.774,0:38:14.758 whether we've got one of those[br]odd ones that looked a little 0:38:14.758,0:38:17.082 bit flat. A kink in the curve.