WEBVTT
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let's find the absolute minimum and absolute maximum values as
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a function F. F. T. Will t
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minus cubic root of tea on the closed interval from
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negative 1 to 4 first. We know that F
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. Is continuous on that close interval. And for
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that reason we know it attains its extreme values over
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that interval. And we know more than that that
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the extreme bodies are attained either at the end points
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of the interval or at critical numbers of F.
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So we gotta find the critical numbers numbers of F
11
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. For that to calculate first the first solar derivative
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of F. And that's one minus uh the relative
13
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of cubic root of T. Yes. If we
14
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write that as T. To the one third is
15
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one third times T to the one third minus one
16
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And that is 1-1 3rd times t. to
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the-2 certs. Yeah. Yeah, He did
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the negative 2/3 in that's equal 21 minus 1/3 times
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T. To the two thirds. And that's discretion
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for the first derivative of F. And this derivative
21
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At zero does not exist. It's not defined there
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. Now Syria is a point in the domain of
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F. in fact effort 0, 0. And
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because Syria is in the domain of the function and
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the relative of F does not exist exist at that
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point. We can say that uh equals zero is
27
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a critical number of earth by definition. Yeah.
28
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Now we got to solve the equations after a difficult
29
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zero and that's equivalent to one minus 1/3 T.
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To the two thirds equal to zero. And that's
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the same as three T to the 2 3rd Equal
32
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one. And in fact this is the same as
33
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T to the 2/3 equal 1/3. In fact we
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are we got to consider all the way through this
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equivalence is here, we got to consider T different
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from zero in order to write this expression like this
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. Okay, so we know that this is the
38
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case because the consider was already treated separately here above
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where we say that there is no derivative there and
40
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for that reason and the point being in the domain
41
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we know that that value of equal serious a critical
42
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number of deaths. We have this equation here,
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we can say that this is the same to T
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square equal one third To the to the 3rd power
45
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. And here we have two solutions to the equal
46
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, more or less, one third to the third
47
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. The square root of that of course let's say
48
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. And that is more or less, we can
49
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write this as one third to the three halves and
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we have two values and the two values here let's
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say this about in order to have an idea of
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zero point. Um See your point so and 19
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245, That's T to the 1/3, That's 1/3
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to the three hats. And both fats are in
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the interval from 21 to 4. So both of
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them had to be considered. So we have three
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critical points. Has three critical numbers In 91 4
58
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. Mhm. Yeah. Which are t equal-1
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3rd to the three House T equals zero and t
60
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equal one third to the three house. That's the
61
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three critical numbers of F in the interim negative 14
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. And now we um Got to evaluate the function
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at those critical numbers and at the end points of
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the interval-14. And we get the following first
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at the endpoints effort negative one is equal to negative
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one minus Nettie 1- Cuba group of native warn
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It is Native one Cuba group of 91 is 91
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And with his minus here in front of the cubic
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root is Plus one. That is serious. Okay
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, so now F at four, which is the
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writing point of the interval is four minus cubic root
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four. And this is about in order to have
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an idea of the value two point 41 259 8948
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. And then we evil wait three critical numbers F
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at negative one third to the three house, this
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one here. Mhm. Yes, negative one third
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to the three house minus one third to the three
78
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house to the one third because the cubic root of
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the number is the same as Putting into the power
80
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of 1/3 and this is the same as negative one
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third to the three house plus. Because here I
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made I forget the sign. Sorry here it's negative
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here. Inside here in front of the branches.
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Is here is this negative of the formula of the
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function. Okay. And then we have the cubic
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root that is racing to the one third. The
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number where we are evaluating which is negative one third
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to the three house of. I forget this negative
89
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sign here. Well. And now because cable group
90
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we have the result is negative with this negative here
91
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out, his plus is a plus sign result get
92
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plus one third And the power end up being 1/2
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. Yeah. Okay so this amount this number here
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, Yes mm 0 0.38 uh 49. And now
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f at zero which is the other critical number is
96
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zero. And if at 1/3 two D three house
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is one third to the three house minus the cubic
98
00:09:03.350 --> 00:09:05.360 A:middle L:90%
root. That is raised to the one third.
99
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The number where we are evaluating the function that is
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1/3 to the three House and that becomes 1 3rd
101
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To the three house-1 3rd to the One House
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. And that's about zero 3849. So from this
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Values here, from these five values, we got
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to find the maximum and the minimum values and functions
105
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we know that these are the maximum value is 2.41
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. Uh The Writing.4. Yeah and the minimum
107
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value is yeah, This one here negative 0.3849.
108
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About that at 1/3 to the three house. So
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with that we can say the answer to the problem
110
00:10:07.840 --> 00:10:22.490 A:middle L:90%
. Mhm. Then the absolute minimum value of F
111
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in the interval from 91-4 is um 2.41 and four
112
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is 4 cubic root of four which is about that's
113
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the exact value. But to have an idea of
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what value it is, it's about Uh 2.41 2598
115
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. To find an 8948 in that absolute minimum value
116
00:11:16.240 --> 00:11:20.419 A:middle L:90%
. Sorry, I make stops or I may stop
117
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the answers here because I'm talking about the minimum,
118
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I wanted to say. Um The minimum value,
119
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we set this one here. So I made a
120
00:11:39.440 --> 00:11:45.860 A:middle L:90%
mistake here. Sorry. The minimum value of the
121
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function is negative. Zero point is one third of
122
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the three house, 1/3 to the three House-1
123
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3rd to the one house, which is about which
124
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is what-0.3849. That is correct. Which of
125
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course At 1/3 to the three health, which is
126
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in fact one of the critical numbers at the critical
127
00:12:31.039 --> 00:12:39.490 A:middle L:90%
number To equal 1/3 to the three house. And
128
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this in fact is about 0.19 245. And they
129
00:12:56.509 --> 00:13:05.409 A:middle L:90%
are now the absolute minimum maximum value of F In
130
00:13:05.409 --> 00:13:13.100 A:middle L:90%
the interval 94, 1 is now it's a value
131
00:13:13.100 --> 00:13:16.610 A:middle L:90%
. It said before is the maximum 2.41. That
132
00:13:16.610 --> 00:13:31.360 A:middle L:90%
is four minus cubic group before is four minus cubic
133
00:13:31.360 --> 00:13:50.360 A:middle L:90%
group of four, which is about 2.141 259 8948
134
00:13:54.539 --> 00:14:03.210 A:middle L:90%
. And that value of course at uh the writing
135
00:14:03.210 --> 00:14:20.950 A:middle L:90%
point of the interval to equal for so, Mhm
136
00:14:20.440 --> 00:14:22.970 A:middle L:90%
. Mhm. Your final answer, which is this
137
00:14:22.970 --> 00:14:28.919 A:middle L:90%
one here? This sentence here says that we have
138
00:14:28.919 --> 00:14:35.840 A:middle L:90%
a natural max actual minimum value of the function of
139
00:14:35.850 --> 00:14:37.860 A:middle L:90%
one third to the three house minus one third to
140
00:14:37.860 --> 00:14:43.429 A:middle L:90%
one half. And that's about 19 0.3849. And
141
00:14:43.429 --> 00:14:48.059 A:middle L:90%
that of course at the critical number 133 House,
142
00:14:48.940 --> 00:14:52.559 A:middle L:90%
which is about 0.19245. And the answer of maximum
143
00:14:52.559 --> 00:14:56.519 A:middle L:90%
value of half in the close interval is four minus
144
00:14:56.519 --> 00:15:07.049 A:middle L:90%
cuba group four, which is about two point 41259484848948
145
00:15:07.059 --> 00:15:07.970 A:middle L:90%
. And that of course at the right end point
146
00:15:07.970 --> 00:15:09.970 A:middle L:90%
of the interval t equals four. So we have
147
00:15:09.980 --> 00:15:15.240 A:middle L:90%
the exact values, extreme violence of the function and
148
00:15:15.240 --> 00:15:20.029 A:middle L:90%
the except values in the interval negative one for where
149
00:15:20.029 --> 00:15:20.899 A:middle L:90%
they occur