WEBVTT 00:00:01.500 --> 00:00:08.105 We now turn our attention to a very interesting application of the second derivative. It's called the 00:00:08.107 --> 00:00:17.200 law of diminishing returns. Let's take a look at this graph. This is a third degree polynomial function. 00:00:17.202 --> 00:00:35.800 This can often be used to model where the x-axis represents the number of dollars spent on advertising. 00:00:35.802 --> 00:00:42.800 So that would be x, and that's a little different. We're used to x usually being a quantity or a number of units sold, 00:00:42.802 --> 00:00:52.801 but here it's dollars spent on advertising. And then the y-axis is going to represent the number of units sold, 00:00:52.803 --> 00:01:09.401 so I'm going to label it N of x for number of units, and then here I'm going to write N of x, number of units sold. 00:01:09.403 --> 00:01:16.200 Now when we model with a function we have what are called real-world restrictions, and they are 00:01:16.202 --> 00:01:23.300 generally pretty obvious. You don't spend negative dollars advertising, so the function is not going 00:01:23.302 --> 00:01:32.501 to be defined here, and the other thing that happens is the model usually cuts off right about here 00:01:32.503 --> 00:01:40.300 at this relative max because as you continue to spend money you hope that your sales will not go down. 00:01:40.302 --> 00:01:46.800 So we're going to have a model that's going to be given to us that's going to have a restricted domain. 00:01:46.802 --> 00:01:57.301 There will be N, some positive value here, and then another larger value. If you realize when you work 00:01:57.303 --> 00:02:03.500 with a function that models number of units sold based on the dollars spent on advertising that this 00:02:03.502 --> 00:02:12.000 is the shape that you indeed get, you can see that right here we have an inflection point, where the 00:02:12.002 --> 00:02:19.601 concavity changes. Roughly over this interval the function is concave up, and then over this interval 00:02:19.603 --> 00:02:26.001 it's concave down. So this is an inflection point which means it is a place where the second derivative 00:02:26.003 --> 00:02:33.100 is either 0 or not defined and the concavity changes. So if you can see that your model matches this, 00:02:33.102 --> 00:02:41.201 you can then draw a very interesting conclusion. What happens is this. As you continue to spend dollars 00:02:41.203 --> 00:02:50.401 on advertising over this entire domain of this function, you can see that the number of units increases. 00:02:50.403 --> 00:02:58.198 This function is increasing over the entire domain, but the rate of change of increase, which would be 00:02:58.200 --> 00:03:06.501 the second derivative, is positive here and negative here. So what that means is when you spend 00:03:06.503 --> 00:03:13.400 money beyond this point, you being the company spending the money on advertising. When you 00:03:13.402 --> 00:03:23.001 spend money beyond this, your sales continue to go up, but the rate of change of sales goes down. 00:03:23.003 --> 00:03:30.699 So generally it is assumed that an investment in advertising beyond this inflection point is not 00:03:30.701 --> 00:03:37.100 a good use of capital, so this turns out to be sort of the perfect amount of money to spend on 00:03:37.102 --> 00:03:43.100 on advertising to increase sales. And how do we find it? We simply use the second derivative test 00:03:43.102 --> 00:03:49.000 and check for an inflection point. We check to make sure the concavity is up and then down. 00:03:49.002 --> 00:03:57.700 Alright, let's take a look at an example of what we would do here. 00:03:57.702 --> 00:04:04.800 The marketing research department of a large recording company has analyzed data on the number of 00:04:04.802 --> 00:04:12.000 CDs that sold after spending money on advertising. Now if you think about this, think about money 00:04:12.002 --> 00:04:18.700 spent on advertising for CDs, and what happens here is generally the more money that's spent, the more 00:04:18.702 --> 00:04:24.701 CDs are sold. But you can't just spend an unlimited amount of money, so you look for that point of 00:04:24.703 --> 00:04:33.200 diminishing returns. Let's see what they came up with. Well, the research people came up with a model. 00:04:33.202 --> 00:04:40.500 They have a third degree polynomial function that models the number of dollars spent on advertising 00:04:40.502 --> 00:04:49.000 as x, and then the model is N of x, which would be the number of CDs sold. This model is increasing 00:04:49.002 --> 00:04:57.201 and therefore only defined over the interval from 1 to 119. Notice that that is thousands of dollars, not 00:04:57.203 --> 00:05:03.900 just dollars, spent on advertising. The question is how much should they spend on advertising in order 00:05:03.902 --> 00:05:12.401 to get the greatest return for the dollar spent? Ok, well let's take a look. We're looking for an inflection 00:05:12.403 --> 00:05:26.901 point between 1 and 119 thousand dollars. Let's see if we can find one. N of x is negative 1/3 x cubed 00:05:26.903 --> 00:05:42.300 plus 60 x squared minus 110x plus 5200 CDs. Now assuming that this model is correct over the domain 00:05:42.302 --> 00:05:58.498 from 1 to 119, we can take the first derivative. Negative x squared plus 120x minus 110, and then take 00:05:58.500 --> 00:06:03.203 the second derivative and look for inflection points. Now when you do this, what you have to be careful 00:06:03.205 --> 00:06:09.100 about is, if you have been practicing the first derivative test for relative extrema and the second 00:06:09.102 --> 00:06:15.400 derivative test for inflection points, your habit here is to just put equals 0 because you're looking for 00:06:15.402 --> 00:06:20.598 critical values of the first derivative which may be relative extrema. We're not interested in that. 00:06:20.600 --> 00:06:25.599 We know that the function is increasing over this interval. We know there are no extrema. We're 00:06:25.601 --> 00:06:34.301 looking for an inflection point, so we just go straight to the second derivative. So we get negative 2x 00:06:34.303 --> 00:06:43.201 plus 120. Now we set this equal to 0 because we're looking for critical values of the second derivative. 00:06:43.203 --> 00:06:54.900 So what do we have? Well, we have negative 2x equals negative 120, and so x is 60, which would be in 00:06:54.902 --> 00:07:01.403 the interval, and it would be 60 thousand dollars. But we have to be sure that we have exactly the 00:07:01.405 --> 00:07:07.703 right concavity here to be sure that this is actually a point of diminishing returns. So we're going to 00:07:07.705 --> 00:07:18.700 do a little sign chart here of the second derivative. Now we only have the function defined from 1. 00:07:18.702 --> 00:07:25.600 We can't really have an arrow here. This is actually 119, and our critical value and our only 00:07:25.602 --> 00:07:37.501 partition number is 60. We pick a test value between 1 and 60, and we go to the second derivative. 00:07:37.503 --> 00:07:45.500 Alright, let's pick anything in here. Let's pick 2. When we put 2 into here, we get negative 4 plus 120, 00:07:45.502 --> 00:07:57.100 and so the sign is going to be positive, which means the function is concave up, the function N. 00:07:57.102 --> 00:08:04.301 And then we pick something in here. We could pick 61. Let's pick maybe 80. It's easy to work with. 00:08:04.303 --> 00:08:14.301 We put 80 into the second derivative. So we get negative 2 times 80, negative 160, plus 120 is negative 40. 00:08:14.303 --> 00:08:20.400 Second derivative is negative. Function is concave down. So what we roughly have over this interval, 00:08:20.402 --> 00:08:25.300 we already know it's increasing, so we have increasing and concave up and then we still have 00:08:25.302 --> 00:08:31.300 increasing and concave down. So the shape is going to look like this, and then like this, 00:08:31.302 --> 00:08:37.400 over the interval from 1 to 119. This matches our model. We found an inflection point. We found the 00:08:37.402 --> 00:08:43.701 point of diminishing returns. So how much should they spend on advertising the CDs? 00:08:43.703 --> 00:08:51.300 60 thousand dollars. Now they could spend more, and their sales would go up, but the rate of return 00:08:51.302 --> 00:08:57.900 would go down. And so this would be the best investment of capital on sales. Now you may 00:08:57.902 --> 00:09:04.500 ask yourself at this point, that's really nice but where did this come from? Well, this comes from 00:09:04.502 --> 00:09:11.900 a model that has been analyzed over time. Records are kept, sales, dollars spent on advertising 00:09:11.902 --> 00:09:17.101 in relation to sales, and then with a simple device like a graphing calculator, you can take 00:09:17.103 --> 00:09:24.000 these points and you can get a model that's good over a certain restricted domain.