WEBVTT 00:00:13.069 --> 00:00:14.868 Hello, Dillon here. 00:00:14.868 --> 00:00:24.584 College Algebra students, we are finally finishing up Section 3.2. 00:00:25.233 --> 00:00:26.402 Here we go. 00:00:26.402 --> 00:00:29.419 There is one last little topic we have to talk about. 00:00:29.419 --> 00:00:37.468 So, some important vocabulary, especially if you're going to take more mathematics after this 00:00:37.468 --> 00:00:43.485 is the idea of extreme values: local minimums and maximums. 00:00:43.485 --> 00:00:49.072 The technical term is local maxima, local minima. 00:00:50.904 --> 00:00:54.452 These are exactly what they sound like… 00:00:54.452 --> 00:00:57.885 in a very local sense. 00:00:58.168 --> 00:01:02.283 And if you focus your attention on a certain part of the polynomial, 00:01:02.283 --> 00:01:05.401 the given point can be a maximum or minimum. 00:01:05.401 --> 00:01:08.901 Here's a value: x equals ‘a.’ 00:01:10.591 --> 00:01:15.099 The technical way of describing this in math is a “neighborhood of a.” 00:01:15.099 --> 00:01:16.638 What does that mean? 00:01:16.638 --> 00:01:18.369 It just means an open interval around it. 00:01:18.369 --> 00:01:25.067 If we just concentrate on this section of the graph, that is a local maximum. 00:01:25.067 --> 00:01:29.300 Why? Because that y value is greater than everything close to it. 00:01:29.300 --> 00:01:35.552 In a similar way, if we concentrate around ‘b,’ again, some small neighborhood around it, 00:01:35.552 --> 00:01:41.751 in that open interval around ‘b,’ that is a local minimum. 00:01:41.751 --> 00:01:45.236 In this case, the polynomial that we're looking at 00:01:45.236 --> 00:01:49.902 has a local maximum and a local minimum. 00:01:50.285 --> 00:01:54.452 Now, keep in mind, this would be an odd-degree polynomial. 00:01:54.452 --> 00:02:01.752 It doesn't have an overall— what’s called a global maximum— because this goes forever in— 00:02:01.752 --> 00:02:07.236 this y value, as x goes to positive infinity, y goes to positive infinity. 00:02:07.236 --> 00:02:09.451 Do you remember that? 00:02:09.934 --> 00:02:14.135 Y goes to positive infinity as x goes to positive infinity. 00:02:14.135 --> 00:02:17.191 This point is a local maximum. 00:02:17.191 --> 00:02:18.485 Listen to the vocabulary. 00:02:18.485 --> 00:02:22.367 We say: “there is a local maximum at x equals a.” 00:02:22.367 --> 00:02:28.840 The maximum value at that local maximum is y equals f of a [f(a)]. 00:02:29.441 --> 00:02:33.574 We say “there's a local minimum at x equals b”; that minimum value, 00:02:33.574 --> 00:02:38.958 that local minimum value is f of b [f(b)]. 00:02:39.940 --> 00:02:44.775 All right, and there's no overall global minimum 00:02:44.775 --> 00:02:50.924 because y goes to negative infinity as x goes to negative infinity. 00:02:55.234 --> 00:02:58.405 Here's an important idea. 00:02:58.497 --> 00:03:01.230 If we have a polynomial of degree n, 00:03:01.230 --> 00:03:07.408 then the graph of P has at most n minus 1 local extreme values. 00:03:07.408 --> 00:03:12.457 So, the n they're referring to is the degree of the polynomial. 00:03:12.457 --> 00:03:15.407 Remember, we like these arranged in descending order. 00:03:15.407 --> 00:03:19.693 This is the leading term, that determines the degree of the polynomial. 00:03:19.693 --> 00:03:23.043 If it's degree n, then it can have at most, 00:03:23.043 --> 00:03:27.943 one less than that degree of local extreme values. 00:03:27.943 --> 00:03:31.860 “Extrema” is the plural for extreme values. 00:03:33.244 --> 00:03:36.493 Okay, so we're going to finish this up by just looking at some graphs, 00:03:36.493 --> 00:03:40.194 and this is probably screenshots from a graphing calculator, 00:03:40.194 --> 00:03:42.877 but it's good enough for us to work with. 00:03:42.877 --> 00:03:52.817 Notice first of all, the first one here, the polynomial P-1 of x is fourth degree. 00:03:54.046 --> 00:04:01.746 Okay, and here's its graph, and notice in this case it has no maximum value 00:04:01.746 --> 00:04:06.530 because our end behavior is to positive infinity on both sides. 00:04:06.530 --> 00:04:10.229 It does have an overall minimum value, this local minimum. 00:04:10.229 --> 00:04:13.679 It's a little hard to tell what it is, but it doesn't matter. 00:04:13.679 --> 00:04:15.479 What we're just looking at is the number. 00:04:15.479 --> 00:04:16.729 It has a local minimum here, 00:04:16.729 --> 00:04:17.995 a local minimum here, 00:04:17.995 --> 00:04:19.812 and then a local maximum. 00:04:20.062 --> 00:04:27.796 Notice, there are three extrema (that's the plural for extreme values), and the degree is 4. 00:04:30.113 --> 00:04:32.145 Not a coincidence. 00:04:32.412 --> 00:04:34.311 Here we have a fifth-degree polynomial, 00:04:34.311 --> 00:04:37.962 that's what the graph would look like, it’s an odd degree. 00:04:37.962 --> 00:04:44.923 Notice this end behavior is consistent with what we've learned earlier in the section. 00:04:44.923 --> 00:04:48.507 Then we have [counting] 1, 2, 3, 4 extreme values. 00:04:48.507 --> 00:04:52.007 Notice that's one less than the degree. 00:04:52.007 --> 00:05:00.776 The theorem up here says that it can have at most n minus 1 local extrema. 00:05:00.776 --> 00:05:03.026 That doesn't mean it has to have them, okay? 00:05:03.026 --> 00:05:05.844 Now in these two examples, it matched exactly: 00:05:05.844 --> 00:05:07.526 fourth degree/three extrema, 00:05:07.526 --> 00:05:10.110 fifth degree/four extrema. 00:05:10.110 --> 00:05:13.009 The point is it can't have more than four. 00:05:13.009 --> 00:05:16.525 Here's exactly what I'm talking about. 00:05:16.525 --> 00:05:26.159 In this last example, so look: fourth degree, it can have at most three (n-1)— 00:05:26.159 --> 00:05:28.309 three extreme values. 00:05:28.309 --> 00:05:30.993 But in fact, this polynomial only has one. 00:05:30.993 --> 00:05:34.209 That's still consistent with the theorem that we just saw. 00:05:34.209 --> 00:05:35.650 This just helps us— 00:05:35.650 --> 00:05:41.526 it’s an additional tool for us to use when we're sketching a graph. 00:05:42.236 --> 00:05:45.475 Okay, wow, that's it. 00:05:45.706 --> 00:05:49.306 We finally finished Section 3.2. 00:05:49.306 --> 00:05:51.939 On to 3.3, bye.