0:00:00.500,0:00:04.270 What I have here in yellow is[br]the graph of y equals f of x. 0:00:04.270,0:00:06.350 Then here in this[br]mauve color I've 0:00:06.350,0:00:09.310 graphed y is equal to[br]the derivative of f 0:00:09.310,0:00:10.870 is f prime of x. 0:00:10.870,0:00:12.580 And then here in[br]blue, I've graphed 0:00:12.580,0:00:15.740 y is equal to the second[br]derivative of our function. 0:00:15.740,0:00:18.380 So this is the[br]derivative of this, 0:00:18.380,0:00:21.550 of the first derivative[br]right over there. 0:00:21.550,0:00:23.830 And we've already seen[br]examples of how can we 0:00:23.830,0:00:25.345 identify minimum[br]and maximum points. 0:00:25.345,0:00:27.220 Obviously if we have[br]the graph in front of us 0:00:27.220,0:00:29.850 it's not hard for a[br]human brain to identify 0:00:29.850,0:00:31.950 this as a local maximum point. 0:00:31.950,0:00:34.530 The function might take[br]on higher values later on. 0:00:34.530,0:00:37.540 And to identify this as[br]a local minimum point. 0:00:37.540,0:00:40.242 The function might take on[br]the lower values later on. 0:00:40.242,0:00:42.700 But we saw, even if we don't[br]have the graph in front of us, 0:00:42.700,0:00:45.347 if we were able to take the[br]derivative of the function 0:00:45.347,0:00:47.180 we might-- or even if[br]we're not able to take 0:00:47.180,0:00:48.638 the derivative of[br]the function-- we 0:00:48.638,0:00:51.501 might be able to identify these[br]points as minimum or maximum. 0:00:51.501,0:00:53.000 The way that we did[br]it, we said, OK, 0:00:53.000,0:00:55.070 what are the critical[br]points for this function? 0:00:55.070,0:00:58.360 Well, critical points are[br]where the function's derivative 0:00:58.360,0:01:00.099 is either undefined or 0. 0:01:00.099,0:01:01.515 This is the[br]function's derivative. 0:01:01.515,0:01:04.170 It is 0 here and here. 0:01:04.170,0:01:05.810 So we would call[br]those critical points. 0:01:05.810,0:01:08.530 And I don't see[br]any points at which 0:01:08.530,0:01:10.590 the derivative is[br]undefined just yet. 0:01:10.590,0:01:15.590 So we would call here[br]and here critical points. 0:01:15.590,0:01:19.740 So these are candidate points[br]at which our function might 0:01:19.740,0:01:21.800 take on a minimum[br]or a maximum value. 0:01:21.800,0:01:23.550 And the way that we[br]figured out whether it 0:01:23.550,0:01:25.220 was a minimum or[br]a maximum value is 0:01:25.220,0:01:29.320 to look at the behavior of the[br]derivative around that point. 0:01:29.320,0:01:36.320 And over here we saw the[br]derivative is positive 0:01:36.320,0:01:37.715 as we approach that point. 0:01:41.210,0:01:42.800 And then it becomes negative. 0:01:42.800,0:01:44.800 It goes from being[br]positive to negative 0:01:44.800,0:01:46.500 as we cross that point. 0:01:46.500,0:01:48.760 Which means that the[br]function was increasing. 0:01:48.760,0:01:50.679 If the derivative[br]is positive, that 0:01:50.679,0:01:52.970 means that the function was[br]increasing as we approached 0:01:52.970,0:01:56.310 that point, and then decreasing[br]as we leave that point, which 0:01:56.310,0:01:58.544 is a pretty good way[br]to think about this 0:01:58.544,0:01:59.460 being a maximum point. 0:01:59.460,0:02:00.765 If we're increasing[br]as we approach it 0:02:00.765,0:02:02.630 and decreasing as we[br]leave it, then this 0:02:02.630,0:02:06.490 is definitely going[br]to be a maximum point. 0:02:06.490,0:02:09.330 Similarly, right[br]over here we see 0:02:09.330,0:02:14.630 that the derivative is negative[br]as we approach the point, which 0:02:14.630,0:02:17.420 means that the[br]function is decreasing. 0:02:17.420,0:02:19.780 And we see that the[br]derivative is positive 0:02:19.780,0:02:20.940 as we exit that point. 0:02:20.940,0:02:22.710 We go from having a[br]negative derivative 0:02:22.710,0:02:24.530 to a positive[br]derivative, which means 0:02:24.530,0:02:27.930 the function goes from[br]decreasing to increasing 0:02:27.930,0:02:30.680 right around that point, which[br]is a pretty good indication, 0:02:30.680,0:02:33.590 or that is the indication,[br]that this critical point is 0:02:33.590,0:02:38.710 a point at which the function[br]takes on a minimum value. 0:02:38.710,0:02:41.150 What I want to do[br]now is extend things 0:02:41.150,0:02:43.270 by using the idea of concavity. 0:02:46.365,0:02:47.740 And I know I'm[br]mispronouncing it. 0:02:47.740,0:02:49.740 Maybe it's concavity. 0:02:49.740,0:02:52.630 But thinking about[br]concavity, we could 0:02:52.630,0:02:55.380 start to look at the second[br]derivative rather than kind 0:02:55.380,0:02:57.990 of seeing just this transition[br]to think about whether this 0:02:57.990,0:03:01.280 is a minimum or a maximum point. 0:03:01.280,0:03:03.300 So let's think about[br]what's happening 0:03:03.300,0:03:06.030 in this first region,[br]this part of the curve 0:03:06.030,0:03:09.540 up here where it looks like[br]a arc where it's opening 0:03:09.540,0:03:11.290 downward, where it[br]looks kind of like an A 0:03:11.290,0:03:13.716 without the cross beam[br]or an upside down U. 0:03:13.716,0:03:15.090 And then we'll[br]think about what's 0:03:15.090,0:03:19.950 happening in this kind of upward[br]opening U part of the curve. 0:03:19.950,0:03:21.790 So over this first[br]interval, right 0:03:21.790,0:03:23.700 over here, if we start[br]over here the slope 0:03:23.700,0:03:26.484 is very-- actually let me do[br]it in a-- actually I'll do it 0:03:26.484,0:03:27.900 in that same color,[br]because that's 0:03:27.900,0:03:30.210 the same color I used for[br]the actual derivative. 0:03:30.210,0:03:33.210 The slope is very positive. 0:03:33.210,0:03:36.640 Then it becomes less positive. 0:03:36.640,0:03:39.790 Then it becomes[br]even less positive. 0:03:39.790,0:03:42.840 It eventually gets to 0. 0:03:42.840,0:03:44.160 Then it keeps decreasing. 0:03:44.160,0:03:47.260 Now it becomes slightly[br]negative, slightly negative, 0:03:47.260,0:03:49.250 then it becomes[br]even more negative, 0:03:49.250,0:03:51.200 then it becomes[br]even more negative. 0:03:51.200,0:03:56.409 And then it looks like it stops[br]decreasing right around there. 0:03:56.409,0:03:58.450 So the slope stops decreasing[br]right around there. 0:03:58.450,0:03:59.380 And you see that[br]in the derivative. 0:03:59.380,0:04:01.380 The slope is decreasing,[br]decreasing, decreasing, 0:04:01.380,0:04:04.970 decreasing until that point,[br]and then it starts to increase. 0:04:04.970,0:04:10.920 So this entire section[br]right over here, 0:04:10.920,0:04:12.575 the slope is decreasing. 0:04:18.550,0:04:21.950 And you see it right over here[br]when we take the derivative. 0:04:21.950,0:04:26.070 The derivative right over[br]here, over this entire interval 0:04:26.070,0:04:27.287 is decreasing. 0:04:27.287,0:04:29.620 And we also see that when we[br]take the second derivative. 0:04:29.620,0:04:31.720 If the derivative[br]is decreasing, that 0:04:31.720,0:04:33.750 means that the second[br]derivative, the derivative 0:04:33.750,0:04:35.050 of the derivative, is negative. 0:04:35.050,0:04:38.320 And we see that that[br]is indeed the case. 0:04:38.320,0:04:43.190 Over this entire interval,[br]the second derivative 0:04:43.190,0:04:45.821 is indeed negative. 0:04:45.821,0:04:47.570 Now what happens as[br]we start to transition 0:04:47.570,0:04:51.250 to this upward opening[br]U part of the curve? 0:04:51.250,0:04:54.150 Well, here the derivative[br]is reasonably negative. 0:04:54.150,0:04:56.250 It's reasonably[br]negative right there. 0:04:56.250,0:04:58.570 But then it's still[br]negative, but it 0:04:58.570,0:05:01.850 becomes less negative and less[br]negative and less negative, 0:05:01.850,0:05:04.670 less negative and less[br]negative, and less negative. 0:05:04.670,0:05:05.980 Then it becomes 0. 0:05:05.980,0:05:08.000 It becomes 0 right over here. 0:05:08.000,0:05:11.170 And then it becomes more[br]and more and more positive. 0:05:11.170,0:05:12.890 And you see that[br]right over here. 0:05:12.890,0:05:16.720 So over this entire interval,[br]the slope or the derivative 0:05:16.720,0:05:18.430 is increasing. 0:05:18.430,0:05:24.745 So the slope is increasing. 0:05:24.745,0:05:25.870 And you see this over here. 0:05:25.870,0:05:27.560 Over here the slope is 0. 0:05:27.560,0:05:29.950 The slope of the[br]derivative is 0. 0:05:29.950,0:05:33.220 The derivative itself isn't[br]changing right at this moment. 0:05:33.220,0:05:37.089 And then you see that[br]the slope is increasing. 0:05:37.089,0:05:38.630 And once again, we[br]can visualize that 0:05:38.630,0:05:40.900 on the second derivative, the[br]derivative of the derivative. 0:05:40.900,0:05:42.530 If the derivative[br]is increasing, that 0:05:42.530,0:05:44.680 means the derivative of[br]that must be positive. 0:05:44.680,0:05:49.240 And it is indeed the case that[br]the derivative is positive. 0:05:49.240,0:05:53.320 And we have a word for[br]this downward opening 0:05:53.320,0:05:57.640 U and this upward opening U.[br]We call this concave downwards. 0:06:02.120,0:06:03.560 Let me make this clear. 0:06:03.560,0:06:07.920 Concave downwards. 0:06:07.920,0:06:10.030 And we call this[br]concave upwards. 0:06:13.400,0:06:15.330 So let's review[br]how we can identify 0:06:15.330,0:06:18.670 concave downward intervals[br]and concave upwards intervals. 0:06:18.670,0:06:25.840 So if we're talking[br]about concave downwards, 0:06:25.840,0:06:27.780 we see several things. 0:06:27.780,0:06:29.340 We see that the[br]slope is decreasing. 0:06:37.590,0:06:41.430 Which is another[br]way of saying that f 0:06:41.430,0:06:47.290 prime of x is decreasing. 0:06:51.720,0:06:54.840 Which is another way of saying[br]that the second derivative must 0:06:54.840,0:06:55.590 be negative. 0:06:55.590,0:06:58.090 If the first derivative[br]is decreasing, 0:06:58.090,0:07:00.460 the second derivative[br]must be negative, 0:07:00.460,0:07:03.350 which is another way of saying[br]that the second derivative 0:07:03.350,0:07:08.160 over that interval[br]must be negative. 0:07:08.160,0:07:11.010 So if you have a negative[br]second derivative, 0:07:11.010,0:07:14.220 then you are in a concave[br]downward interval. 0:07:14.220,0:07:17.480 Similarly-- I have[br]trouble saying 0:07:17.480,0:07:21.640 that word-- let's think[br]about concave upwards, 0:07:21.640,0:07:25.630 where you have an upward[br]opening U. Concave upwards. 0:07:25.630,0:07:28.510 In these intervals, the[br]slope is increasing. 0:07:28.510,0:07:31.030 We have a negative slope, less[br]negative, less negative, 0, 0:07:31.030,0:07:34.040 positive, more positive, more[br]positive, even more positive. 0:07:34.040,0:07:37.780 So slope is increasing. 0:07:43.240,0:07:50.690 Which means that the derivative[br]of the function is increasing. 0:07:50.690,0:07:52.680 And you see that[br]right over here. 0:07:52.680,0:07:56.240 This derivative is[br]increasing in value, 0:07:56.240,0:08:00.150 which means that the second[br]derivative over an interval 0:08:00.150,0:08:03.490 where we are concave upwards[br]must be greater than 0. 0:08:03.490,0:08:05.450 If the second derivative[br]is greater than 0, 0:08:05.450,0:08:06.950 that means that the[br]first derivative 0:08:06.950,0:08:09.310 is increasing, which means[br]that the slope is increasing. 0:08:09.310,0:08:14.630 We are in a concave[br]upward interval. 0:08:14.630,0:08:18.000 Now given all of these[br]definitions that we've just 0:08:18.000,0:08:20.150 given for concave downwards[br]and concave upwards, 0:08:20.150,0:08:22.210 can we come up with[br]another way of identifying 0:08:22.210,0:08:24.820 whether a critical[br]point is a minimum point 0:08:24.820,0:08:26.490 or a maximum point? 0:08:26.490,0:08:28.420 Well, if you have[br]a maximum point, 0:08:28.420,0:08:32.500 if you have a critical[br]point where the function is 0:08:32.500,0:08:36.130 concave downwards, then you're[br]going to be at a maximum point. 0:08:36.130,0:08:38.210 Concave downwards, let's[br]just be clear here, 0:08:38.210,0:08:42.263 means that it's[br]opening down like this. 0:08:42.263,0:08:44.179 And when we're talking[br]about a critical point, 0:08:44.179,0:08:46.304 if we're assuming it's[br]concave downwards over here, 0:08:46.304,0:08:48.784 we're assuming differentiability[br]over this interval. 0:08:48.784,0:08:50.200 And so the critical[br]point is going 0:08:50.200,0:08:52.350 to be one where the slope is 0. 0:08:52.350,0:08:54.810 So it's going to be that[br]point right over there. 0:08:54.810,0:08:56.640 So if you're concave[br]downwards and you 0:08:56.640,0:09:02.250 have a point where f prime of,[br]let's say, a is equal to 0, 0:09:02.250,0:09:04.730 then we have a[br]maximum point at a. 0:09:11.500,0:09:14.210 And similarly, if[br]we're concave upwards, 0:09:14.210,0:09:17.340 that means that our function[br]looks something like this. 0:09:17.340,0:09:20.210 And if we found a point,[br]obviously a critical point 0:09:20.210,0:09:22.600 could also be where the[br]function is not defined, 0:09:22.600,0:09:24.800 but if we're assuming[br]that our first derivative 0:09:24.800,0:09:26.510 and second derivative[br]is defined here, 0:09:26.510,0:09:28.260 then the critical point[br]is going to be one 0:09:28.260,0:09:31.380 where the first derivative[br]is going to be 0. 0:09:31.380,0:09:35.090 So f prime of a is equal to 0. 0:09:35.090,0:09:38.130 And if f prime of[br]a is equal to 0 0:09:38.130,0:09:40.900 and if we're concave[br]upwards in the interval 0:09:40.900,0:09:44.250 around a, so if the second[br]derivative is greater than 0, 0:09:44.250,0:09:46.010 then it's pretty[br]clear, you see here, 0:09:46.010,0:09:53.610 that we are dealing with[br]a minimum point at a.