WEBVTT
1
00:00:00.040 --> 00:00:01.860 A:middle L:90%
All right, So we're being told to find a
2
00:00:01.860 --> 00:00:04.910 A:middle L:90%
cubic function f of x equals X cubed, possibly
3
00:00:04.910 --> 00:00:07.009 A:middle L:90%
x squared. Plus the 8. 50 that has
4
00:00:07.009 --> 00:00:10.769 A:middle L:90%
a local max of three at X equals negative two
5
00:00:10.769 --> 00:00:14.490 A:middle L:90%
in a local men of 0 64 1. So
6
00:00:14.490 --> 00:00:17.899 A:middle L:90%
that's all the information we're given. So the first
7
00:00:17.899 --> 00:00:20.230 A:middle L:90%
thing we want to do is take the derivative.
8
00:00:20.239 --> 00:00:23.149 A:middle L:90%
So we want to take the derivative of primer back
9
00:00:25.440 --> 00:00:30.059 A:middle L:90%
, and this will give us three a x squared
10
00:00:30.739 --> 00:00:34.299 A:middle L:90%
plus b not B expert going to be to be
11
00:00:34.299 --> 00:00:40.850 A:middle L:90%
xr to be x plus c. And so,
12
00:00:41.439 --> 00:00:44.350 A:middle L:90%
um, the reason I took the derivative is because
13
00:00:44.840 --> 00:00:47.560 A:middle L:90%
we're told some information about the derivative. Believe it
14
00:00:47.560 --> 00:00:49.549 A:middle L:90%
or not, we're told that the local Max,
15
00:00:50.240 --> 00:00:52.200 A:middle L:90%
um, and a local men So what is the
16
00:00:52.200 --> 00:00:55.070 A:middle L:90%
local Max and local men means that there has to
17
00:00:55.070 --> 00:00:59.479 A:middle L:90%
be a point at local maximum ensures that F prime
18
00:00:59.479 --> 00:01:02.689 A:middle L:90%
is equal to zero. So we're told that at
19
00:01:02.689 --> 00:01:06.819 A:middle L:90%
these values, negative two and one, So I
20
00:01:06.829 --> 00:01:11.640 A:middle L:90%
have prime of negative. Coup is equal to f
21
00:01:11.640 --> 00:01:15.219 A:middle L:90%
prime of one because that's when we have a local
22
00:01:15.219 --> 00:01:17.719 A:middle L:90%
max. So when a local massacres, when it
23
00:01:17.719 --> 00:01:19.659 A:middle L:90%
goes like this or in the local ministers like this
24
00:01:19.670 --> 00:01:23.939 A:middle L:90%
. But both have a derivative zero. So we
25
00:01:23.939 --> 00:01:26.069 A:middle L:90%
know that this fact is true. Mm. So
26
00:01:26.069 --> 00:01:27.920 A:middle L:90%
what do we do now? We Well, we
27
00:01:27.920 --> 00:01:33.060 A:middle L:90%
can just plug in our values into X for F
28
00:01:33.060 --> 00:01:36.849 A:middle L:90%
prime. So we get three a, then you
29
00:01:36.849 --> 00:01:40.310 A:middle L:90%
get negative. Two squared. No, no.
30
00:01:40.319 --> 00:01:44.700 A:middle L:90%
Um, no, no, I'm sorry. This
31
00:01:44.700 --> 00:01:49.900 A:middle L:90%
is going to be, uh So what? We're
32
00:01:49.900 --> 00:01:51.790 A:middle L:90%
going? Yeah. So we're going to plug in
33
00:01:51.799 --> 00:01:53.650 A:middle L:90%
. We're gonna plug these numbers into, uh,
34
00:01:55.340 --> 00:01:59.359 A:middle L:90%
X right here. So we're gonna get three a
35
00:02:00.439 --> 00:02:02.659 A:middle L:90%
and the negative two squared, plus, um,
36
00:02:02.670 --> 00:02:07.469 A:middle L:90%
to be and then negative too. And then plus
37
00:02:07.469 --> 00:02:09.919 A:middle L:90%
C and then we're gonna plug in one. So
38
00:02:09.919 --> 00:02:13.090 A:middle L:90%
for one, we just get three a plus two
39
00:02:13.090 --> 00:02:17.199 A:middle L:90%
b plus c. And then once you simplify this
40
00:02:17.199 --> 00:02:23.930 A:middle L:90%
, you get 12 a minus four b equals three
41
00:02:23.930 --> 00:02:27.259 A:middle L:90%
. A push to be because the sea is canceled
42
00:02:27.939 --> 00:02:30.419 A:middle L:90%
. And now we can solve for either air B
43
00:02:30.419 --> 00:02:30.930 A:middle L:90%
. I solve for B in this case. So
44
00:02:30.930 --> 00:02:34.960 A:middle L:90%
he's all for B. You get three have.
45
00:02:35.340 --> 00:02:39.879 A:middle L:90%
Hey, so now we what we do as we
46
00:02:39.889 --> 00:02:44.889 A:middle L:90%
, um we know that f prime of one.
47
00:02:44.889 --> 00:02:46.509 A:middle L:90%
So I'm gonna go ahead and put in red.
48
00:02:46.509 --> 00:02:47.750 A:middle L:90%
So this is our second step. So we know
49
00:02:47.750 --> 00:02:51.159 A:middle L:90%
that f prime of one is equal to zero.
50
00:02:52.939 --> 00:02:54.669 A:middle L:90%
Yeah. And so what we're gonna do is we're
51
00:02:54.669 --> 00:03:00.770 A:middle L:90%
gonna set, um three a X squared, just
52
00:03:00.780 --> 00:03:06.340 A:middle L:90%
to be x plus C equals zero. So equal
53
00:03:06.340 --> 00:03:09.039 A:middle L:90%
zero. And we're gonna plug in one. So
54
00:03:09.039 --> 00:03:10.610 A:middle L:90%
this is going to come out to be, um
55
00:03:10.620 --> 00:03:15.639 A:middle L:90%
, zero. It's equal to three. A plus
56
00:03:15.639 --> 00:03:20.539 A:middle L:90%
to be. Let's see. Well, I'm gonna
57
00:03:20.539 --> 00:03:21.969 A:middle L:90%
bring this on to the next page. I'm gonna
58
00:03:21.969 --> 00:03:23.009 A:middle L:90%
rewrite this so it's gonna be zero. Is he
59
00:03:23.009 --> 00:03:25.090 A:middle L:90%
called. I'm gonna go ahead and continue the red
60
00:03:25.090 --> 00:03:30.659 A:middle L:90%
color to show are different. Step Oops, sorry
61
00:03:32.039 --> 00:03:38.150 A:middle L:90%
. Zero is equal to three. Um, a
62
00:03:39.340 --> 00:03:45.520 A:middle L:90%
plus to be plus c now, reminder. We
63
00:03:45.520 --> 00:03:47.719 A:middle L:90%
saw for B so we can actually plug in or
64
00:03:47.719 --> 00:03:52.659 A:middle L:90%
Newbie. So zero goes three a plus, two
65
00:03:53.240 --> 00:03:59.539 A:middle L:90%
times, 3/2 a and m plus c. And
66
00:03:59.539 --> 00:04:00.050 A:middle L:90%
now what we can do is we can just all
67
00:04:00.050 --> 00:04:04.360 A:middle L:90%
for C galaxy is gonna give us negative six A
68
00:04:04.939 --> 00:04:09.120 A:middle L:90%
. So now I'm gonna circle some of the important
69
00:04:09.129 --> 00:04:12.219 A:middle L:90%
, um, constant. So we have b go
70
00:04:12.219 --> 00:04:14.900 A:middle L:90%
to three half a and see you to go the
71
00:04:14.900 --> 00:04:16.300 A:middle L:90%
negative six a. Now we're gonna move on to
72
00:04:16.300 --> 00:04:21.360 A:middle L:90%
another step, and that is f of negative two
73
00:04:21.370 --> 00:04:24.860 A:middle L:90%
. So what we're going to do so have a
74
00:04:24.860 --> 00:04:30.279 A:middle L:90%
negative too. Now that we know the values of
75
00:04:30.290 --> 00:04:31.360 A:middle L:90%
, um hey, we can actually go ahead and
76
00:04:31.360 --> 00:04:36.170 A:middle L:90%
start plugging this into our original function. I mean
77
00:04:36.170 --> 00:04:38.610 A:middle L:90%
, we know the value of B and C,
78
00:04:38.610 --> 00:04:40.269 A:middle L:90%
so we're gonna go ahead and start plugging this into
79
00:04:40.269 --> 00:04:44.449 A:middle L:90%
our original function. Yeah, so we're going to
80
00:04:44.449 --> 00:04:46.350 A:middle L:90%
get f of negative two. So that's gonna give
81
00:04:46.350 --> 00:04:50.949 A:middle L:90%
us negative tube cube times a and then plus negative
82
00:04:50.959 --> 00:04:57.170 A:middle L:90%
two squared and in times be would be is three
83
00:04:57.170 --> 00:05:03.449 A:middle L:90%
half a and then we're gonna get Plus, this
84
00:05:03.449 --> 00:05:06.930 A:middle L:90%
is gonna be negative two times, see, and
85
00:05:06.930 --> 00:05:10.620 A:middle L:90%
then plus D. And this is equal to three
86
00:05:10.620 --> 00:05:13.269 A:middle L:90%
because this is told to us effort at X equals
87
00:05:13.269 --> 00:05:15.720 A:middle L:90%
negative two. We get three mhm. We're going
88
00:05:15.720 --> 00:05:19.660 A:middle L:90%
to hold on to this value right here and now
89
00:05:19.660 --> 00:05:25.420 A:middle L:90%
we're going to solve for f of one so half
90
00:05:25.420 --> 00:05:30.259 A:middle L:90%
of one. Well, that's going to be a
91
00:05:30.639 --> 00:05:32.750 A:middle L:90%
plus three half a, which is our B.
92
00:05:32.750 --> 00:05:36.230 A:middle L:90%
But now it's three half a rewritten then plus C
93
00:05:36.240 --> 00:05:40.879 A:middle L:90%
and then plus D. And now this can be
94
00:05:40.879 --> 00:05:46.250 A:middle L:90%
rewritten as five half a, um, plus the
95
00:05:46.839 --> 00:05:49.149 A:middle L:90%
plus d and this is all equal to zero.
96
00:05:53.379 --> 00:05:56.560 A:middle L:90%
Then we also know that f of negative two.
97
00:05:58.939 --> 00:06:00.740 A:middle L:90%
You can also, um, So what we can
98
00:06:00.740 --> 00:06:04.560 A:middle L:90%
also do now is we're going to do, um
99
00:06:04.569 --> 00:06:10.250 A:middle L:90%
, f of negative two minus f of one.
100
00:06:10.290 --> 00:06:12.730 A:middle L:90%
And that's gonna give us three minus zero, which
101
00:06:12.730 --> 00:06:16.199 A:middle L:90%
is still equal to three. So f of two
102
00:06:16.209 --> 00:06:21.689 A:middle L:90%
minus f uh, one. This is delicate three
103
00:06:21.699 --> 00:06:26.550 A:middle L:90%
. And then this is equal to negative to a
104
00:06:28.139 --> 00:06:34.639 A:middle L:90%
minus five. Half a managed to see minus c
105
00:06:34.649 --> 00:06:38.959 A:middle L:90%
. So I did some substitution here. Um,
106
00:06:39.939 --> 00:06:42.699 A:middle L:90%
Now we can get another value for See that we
107
00:06:42.699 --> 00:06:46.470 A:middle L:90%
can rewrite to see negative one minus three, half
108
00:06:46.480 --> 00:06:50.980 A:middle L:90%
A. We also know. So now we have
109
00:06:50.980 --> 00:06:54.439 A:middle L:90%
to seize. Now that we have to seize,
110
00:06:54.439 --> 00:06:56.839 A:middle L:90%
we cannot just solve for what number A is.
111
00:06:56.839 --> 00:06:59.689 A:middle L:90%
Remember, we succeed equal to negative six A.
112
00:06:59.699 --> 00:07:01.220 A:middle L:90%
So now we have two versions of C, so
113
00:07:01.220 --> 00:07:04.459 A:middle L:90%
we can say negative. Six a is equal to
114
00:07:04.459 --> 00:07:09.699 A:middle L:90%
negative one minus three half. Okay. And if
115
00:07:09.699 --> 00:07:11.560 A:middle L:90%
you saw for this, you get a you go
116
00:07:11.560 --> 00:07:15.339 A:middle L:90%
to to ninth. Wow. So Oh, that's
117
00:07:15.339 --> 00:07:19.550 A:middle L:90%
great. Um, now we know a So now
118
00:07:19.550 --> 00:07:23.250 A:middle L:90%
that we know A we also know that B is
119
00:07:23.250 --> 00:07:27.779 A:middle L:90%
equal to three half times a so three half and
120
00:07:27.779 --> 00:07:30.769 A:middle L:90%
then we know we just saw for a is 29
121
00:07:30.779 --> 00:07:33.629 A:middle L:90%
to 9. So you get 39 which is just
122
00:07:33.629 --> 00:07:40.550 A:middle L:90%
one third. So now we don't be well,
123
00:07:41.139 --> 00:07:44.079 A:middle L:90%
now that we know what B is and we know
124
00:07:44.079 --> 00:07:46.670 A:middle L:90%
what a is Remember, let's see what's equal to
125
00:07:46.670 --> 00:07:48.399 A:middle L:90%
negative six a. And we know what a s
126
00:07:48.399 --> 00:07:54.750 A:middle L:90%
a negative six time to ninth and that comes out
127
00:07:54.750 --> 00:07:58.430 A:middle L:90%
to be negative for third. So now we have
128
00:07:58.439 --> 00:08:01.839 A:middle L:90%
a B A B N c. And to find
129
00:08:01.839 --> 00:08:03.560 A:middle L:90%
what D is we can There's a There's a number
130
00:08:03.560 --> 00:08:05.779 A:middle L:90%
of ways to do this. I'm just gonna do
131
00:08:05.779 --> 00:08:09.120 A:middle L:90%
the simple one. So everyone which is a B
132
00:08:09.120 --> 00:08:11.959 A:middle L:90%
plus a plus B plus C plus d, which
133
00:08:11.959 --> 00:08:18.110 A:middle L:90%
is gonna be to ninth plus one third and then
134
00:08:18.110 --> 00:08:20.459 A:middle L:90%
she is negative. So it's me minus four third
135
00:08:22.040 --> 00:08:24.430 A:middle L:90%
and then plus D and we know that everyone is
136
00:08:24.430 --> 00:08:26.360 A:middle L:90%
equal to zero. And if you saw for D
137
00:08:26.839 --> 00:08:31.639 A:middle L:90%
, you got 7/9. So now we have salt
138
00:08:31.639 --> 00:08:33.879 A:middle L:90%
for A B and A, B, C and
139
00:08:33.879 --> 00:08:39.269 A:middle L:90%
D. And if you plug in those values into
140
00:08:39.269 --> 00:08:41.879 A:middle L:90%
our original function, I mean to our original form
141
00:08:41.879 --> 00:08:43.259 A:middle L:90%
, that is the cubic function.