1 00:00:00,012 --> 00:00:04,324 [music] Here's kind of an old timey puzzle about numbers. 2 00:00:04,324 --> 00:00:07,797 Suppose you've got 2 numbers which add up to 24. 3 00:00:07,797 --> 00:00:12,707 You've got 2 numbers that sum to 24. How large can their product be? 4 00:00:12,707 --> 00:00:18,231 When you take those 2 numbers that add up to 24, how big of a number can you get 5 00:00:18,231 --> 00:00:24,233 when you multiply them together? This is an optimization problem so let's 6 00:00:24,234 --> 00:00:28,769 draw a picture. Okay, maybe I can't draw a picture in this 7 00:00:28,769 --> 00:00:34,759 problem but at least I can label things. So, instead of just saying two numbers 8 00:00:34,759 --> 00:00:40,889 that sum to 24, let's give them the name. Two numbers, x and y, which sum to, to 24. 9 00:00:40,890 --> 00:00:47,174 And instead of just saying how large can their product be, I can say, how large 10 00:00:47,174 --> 00:00:50,901 can, you know, x times y be? So, what's the goal? 11 00:00:50,901 --> 00:00:57,243 Well, my goal here is really just to maximize the product, so maximize x times 12 00:00:57,243 --> 00:00:59,911 y. And there's a constraint. 13 00:00:59,911 --> 00:01:04,421 The constraint is that x and y have to add up to 24. 14 00:01:04,421 --> 00:01:08,856 So, I'll write that as x plus y equals 24. Rats. 15 00:01:08,856 --> 00:01:15,348 X,y isn't a function of a single variable. I need to rewrite it so it's a function of 16 00:01:15,348 --> 00:01:19,240 x alone. Now, since x plus y is 24, I could rewrite 17 00:01:19,240 --> 00:01:24,671 this as y equals 24 minus x. And consequently, x times y, which is the 18 00:01:24,671 --> 00:01:30,431 thing I'm trying to maximize, I could write that as a function of a single 19 00:01:30,431 --> 00:01:34,113 variable, x times, and instead of y 24 minus x. 20 00:01:34,113 --> 00:01:37,381 So, this is the quantity now that I want to maximize. 21 00:01:37,381 --> 00:01:41,534 Now, I can apply Calculus. Let's differentiate and then find the 22 00:01:41,534 --> 00:01:45,036 critical points. So, here's this function of a single 23 00:01:45,036 --> 00:01:49,947 variable that I'm trying to maximize. My function is x times 24 minus x, that 24 00:01:49,947 --> 00:01:54,284 should be y or this is x times y, this thing I'm trying to maximize. 25 00:01:54,284 --> 00:01:57,021 But if I expand this out, I get 24x minus x squared. 26 00:01:57,021 --> 00:02:01,394 Then, I can easily differentiate this. The derivative of this function is, what's 27 00:02:01,394 --> 00:02:03,142 the derivative of 24x? It's 24. 28 00:02:03,142 --> 00:02:06,072 What's the derivative minus x squared? It's minus 2x. 29 00:02:06,072 --> 00:02:11,017 I'm trying to find critical points, right? Critical points are where the derivative 30 00:02:11,017 --> 00:02:14,280 doesn't exist or where the derivative is equal to 0. 31 00:02:14,280 --> 00:02:17,972 Now, this function's just a polynomial, so it's differentiable everywhere. 32 00:02:17,972 --> 00:02:21,516 So, I don't have to worry about the functions derivative not existing 33 00:02:21,516 --> 00:02:23,941 somewhere. But I do have to find places where the 34 00:02:23,941 --> 00:02:27,306 derivative is equal to zero. So, when is this thing equal to zero? 35 00:02:27,307 --> 00:02:31,171 Well, that's the same as asking, when 24 is 2x? 36 00:02:31,171 --> 00:02:35,751 That's exactly when x is 12. So, here is my critical point. 37 00:02:35,751 --> 00:02:40,888 What sort of point is that? Well, let's think about the s, i, g, n, 38 00:02:40,888 --> 00:02:47,272 the sign of the first derivative. The derivative is positive if x is less 39 00:02:47,272 --> 00:02:52,701 than 12, and the derivative is negative if x is bigger than 12. 40 00:02:52,701 --> 00:02:57,225 What does that mean? Well, that means the function is 41 00:02:57,225 --> 00:03:03,439 increasing for x values less than 12. And the function is decreasing for x 42 00:03:03,439 --> 00:03:08,945 values bigger than 12. So, the function goes up, gets to 12, and 43 00:03:08,945 --> 00:03:14,009 starts going down. What kind of point does that make 12? 44 00:03:14,009 --> 00:03:19,555 Well, that must be a maximum. So, I'll write f of 12 is my maximum 45 00:03:19,555 --> 00:03:22,334 value. So, what do we conclude? 46 00:03:22,334 --> 00:03:27,727 So, here's our conclusion. A 144 is the maximum product of two 47 00:03:27,727 --> 00:03:32,752 numbers which sum to 24. Did we really need the awesome power of 48 00:03:32,752 --> 00:03:35,386 Calculus to solve this problem? No. 49 00:03:35,386 --> 00:03:39,366 For example, you could have used a technique like this. 50 00:03:39,366 --> 00:03:43,751 There's the so-called arithmetic geometric mean inequality. 51 00:03:43,751 --> 00:03:48,156 The AM-GM inequality. And what it tells you is the following. 52 00:03:48,156 --> 00:03:54,460 That for numbers a and b, the arithmetic mean, just a fancy word for the average of 53 00:03:54,460 --> 00:04:00,476 a and b, is bigger than or equal to the geometric mean, which is the square root 54 00:04:00,476 --> 00:04:05,933 of the product of a and b. And this inequality becomes an equality, 55 00:04:05,933 --> 00:04:11,937 if and only if a and b are the same. Now, how can you use something like this? 56 00:04:11,937 --> 00:04:15,467 Well, in our specific case, what do we know? 57 00:04:15,467 --> 00:04:21,647 I know that x plus y is equal to 24 and if these two numbers add to 24, then their 58 00:04:21,647 --> 00:04:25,044 average x plus y over 2 must be equal to 12. 59 00:04:25,044 --> 00:04:31,122 But if this is their average by the AM-GM inequality, this must be bigger than or 60 00:04:31,122 --> 00:04:36,178 equal to the square root of x times y, the geometric mean of x and y. 61 00:04:36,178 --> 00:04:41,469 Now, how does this help us? Well, what if I square both sides, right? 62 00:04:41,469 --> 00:04:45,280 Then I know that 144 is bigger than or equal to x times y. 63 00:04:45,280 --> 00:04:48,236 And that's exactly what I'm trying to figure out. 64 00:04:48,236 --> 00:04:52,904 I'm trying to figure out how big can the product of x and y be, if I know what the 65 00:04:52,904 --> 00:04:55,732 sum is. And what this is telling me, is that the 66 00:04:55,732 --> 00:05:00,294 product can be no bigger than 144, with equality when x and y are both equal. 67 00:05:00,294 --> 00:05:03,274 And that happens when they are both equal to 12. 68 00:05:03,275 --> 00:05:08,452 That you don't actually need Calculus to solve many of the problems that are 69 00:05:08,452 --> 00:05:13,892 assigned in Calculus courses, is perhaps one of the best kept secrets of Calculus 70 00:05:13,892 --> 00:05:17,361 instructors. A result like this AM-GM inequality, 71 00:05:17,361 --> 00:05:20,631 actually solves a ton of optimization problems. 72 00:05:20,631 --> 00:05:25,740 You often don't have to resort to taking derivatives, if you can just apply a 73 00:05:25,740 --> 00:05:32,116 result like this. Calculus is a powerful tool but there are 74 00:05:32,116 --> 00:05:40,955 other ways to attack these problems. Don't be afraid to make use of other 75 00:05:40,955 --> 00:05:42,147 methods.