1
00:00:00,000 --> 00:00:00,660


2
00:00:00,660 --> 00:00:02,940
Let's see if we can use everything we know about

3
00:00:02,940 --> 00:00:06,019
differentiation and concativity, and maximum and

4
00:00:06,019 --> 00:00:09,759
minimum points, and inflection points, to actually graph a

5
00:00:09,759 --> 00:00:12,529
function without using a graphing calculator.

6
00:00:12,529 --> 00:00:16,649
So let's say our function, let's say that f of x is equal

7
00:00:16,649 --> 00:00:27,629
to 3x to the fourth minus 4x to the third plus 2.

8
00:00:27,629 --> 00:00:30,849
And of course, you could always graph a function just by trying

9
00:00:30,850 --> 00:00:33,370
out a bunch of points, but we want to really focus on the

10
00:00:33,369 --> 00:00:35,829
points that are interesting to us, and then just to get the

11
00:00:35,829 --> 00:00:38,239
general shape of the function, especially we want to focus on

12
00:00:38,240 --> 00:00:42,160
the things that we can take out from this function using our

13
00:00:42,159 --> 00:00:44,929
calculus toolkit, or our derivative toolkit.

14
00:00:44,929 --> 00:00:46,939
So the first thing we probably want to do, is figure

15
00:00:46,939 --> 00:00:48,679
out the critical points.

16
00:00:48,679 --> 00:00:54,590
We want to figure out, I'll write here, critical points.

17
00:00:54,590 --> 00:00:58,520
And just as a refresher of what critical points means, it's the

18
00:00:58,520 --> 00:01:02,320
points where the derivative of f of x is 0.

19
00:01:02,320 --> 00:01:06,200
So critical points are f prime of x is either equal to

20
00:01:06,200 --> 00:01:08,460
0, or it's undefined.

21
00:01:08,459 --> 00:01:12,159


22
00:01:12,159 --> 00:01:14,609
This function looks differentiable everywhere, so

23
00:01:14,609 --> 00:01:17,319
the critical points that we worried about are probably,

24
00:01:17,319 --> 00:01:19,919
well, I can tell you, they're definitely just the points

25
00:01:19,920 --> 00:01:22,579
where f prime of x are going to be equal to 0.

26
00:01:22,579 --> 00:01:26,620
This derivative, f prime of x, is going to actually be defined

27
00:01:26,620 --> 00:01:27,840
over the entire domain.

28
00:01:27,840 --> 00:01:31,010
So let's actually write down the derivative right now.

29
00:01:31,010 --> 00:01:36,390
So the derivative of this, f prime of x, this is

30
00:01:36,390 --> 00:01:37,469
pretty straightforward.

31
00:01:37,469 --> 00:01:42,530
The derivative of 3x to the fourth, 4 times 3 is 12,

32
00:01:42,530 --> 00:01:47,420
12x to the, we'll just decrement the 4 by 1, 3.

33
00:01:47,420 --> 00:01:47,750
Right?

34
00:01:47,750 --> 00:01:50,909
You just multiply times the exponent, and then decrease the

35
00:01:50,909 --> 00:01:57,019
new exponent by one, minus 3 times 4 is 12, times x to

36
00:01:57,019 --> 00:01:59,909
the 1 less than 3 is 2.

37
00:01:59,909 --> 00:02:02,989
And then the derivative of a constant, the slope of

38
00:02:02,989 --> 00:02:04,809
a constant, you could almost imagine, is zero.

39
00:02:04,810 --> 00:02:05,600
It's not changing.

40
00:02:05,599 --> 00:02:07,349
A constant, by definition, isn't changing.

41
00:02:07,349 --> 00:02:08,883
So that's f prime of x.

42
00:02:08,883 --> 00:02:11,229
So let's figure out the critical points.

43
00:02:11,229 --> 00:02:13,750
The critical points are where this thing is either going to

44
00:02:13,750 --> 00:02:16,270
be equal to 0, or it's undefined.

45
00:02:16,270 --> 00:02:18,930
Now, I can look over the entire domain of real numbers, and

46
00:02:18,930 --> 00:02:20,610
this thing is defined pretty much anywhere.

47
00:02:20,610 --> 00:02:22,970
I could put any number here, and it's not going to blow up.

48
00:02:22,969 --> 00:02:26,719
It's going to give me an answer to what the function is.

49
00:02:26,719 --> 00:02:28,680
So that it's defined everywhere, so let's

50
00:02:28,680 --> 00:02:30,400
just figure out where it's equal to 0.

51
00:02:30,400 --> 00:02:35,300
So f prime of x is equal to 0.

52
00:02:35,300 --> 00:02:38,670
So let's solve, which x is, let's solve-- I don't have

53
00:02:38,669 --> 00:02:41,129
to rewrite that, I just wrote that.

54
00:02:41,129 --> 00:02:43,569
Let's solve for when this is equal to 0.

55
00:02:43,569 --> 00:02:45,349
And I'll do it in the same color.

56
00:02:45,349 --> 00:02:56,979
So 12x to the third minus 12x squared is equal to 0.

57
00:02:56,979 --> 00:02:59,530
And so let's what we can do to solve this.

58
00:02:59,530 --> 00:03:01,860
We could factor out a 12x.

59
00:03:01,860 --> 00:03:07,830
So if we factor out a 12x, then this term becomes just x,

60
00:03:07,830 --> 00:03:11,450
and then-- actually, let's factor out a 12x squared.

61
00:03:11,449 --> 00:03:12,899
We factor out a 12x squared.

62
00:03:12,900 --> 00:03:15,500
If we divide both of these by 12x squared, this term just

63
00:03:15,500 --> 00:03:19,990
becomes an x, and then minus 12x squared divided by 12x

64
00:03:19,990 --> 00:03:22,650
squared is just 1, is equal to 0.

65
00:03:22,650 --> 00:03:24,680
I just rewrote this top thing like this.

66
00:03:24,680 --> 00:03:25,370
You could go the other way.

67
00:03:25,370 --> 00:03:28,420
If I distributed this 12x squared times this entire

68
00:03:28,419 --> 00:03:31,869
quantity, you would get my derivative right there.

69
00:03:31,870 --> 00:03:36,530
So the reason why did that is because, to solve for 0, or if

70
00:03:36,530 --> 00:03:41,530
I want all of the x's that make this equation equal to 0, I now

71
00:03:41,530 --> 00:03:43,960
have written it in a form where I'm multiplying one

72
00:03:43,960 --> 00:03:46,250
thing by another thing.

73
00:03:46,250 --> 00:03:49,960
And in order for this to be 0, one or both of these

74
00:03:49,960 --> 00:03:51,480
things must be equal to 0.

75
00:03:51,479 --> 00:03:55,919
So 12x squared are equal to 0, which means that x is equal

76
00:03:55,919 --> 00:03:58,750
to 0 will make this quantity equals 0.

77
00:03:58,750 --> 00:04:01,254
And the other thing that would make this quantity 0 is if

78
00:04:01,254 --> 00:04:04,099
x minus 1 is equal to 0.

79
00:04:04,099 --> 00:04:08,049
So x minus 1 is equal to 0 when x is equal to 1.

80
00:04:08,050 --> 00:04:10,510
So these are 2 critical points.

81
00:04:10,509 --> 00:04:14,560
Our 2 critical points are x is equal to 0 and x is equal to 1.

82
00:04:14,560 --> 00:04:18,189
And remember, those are just the points where our first

83
00:04:18,189 --> 00:04:20,050
derivative is equal to 0.

84
00:04:20,050 --> 00:04:22,280
Where the slope is 0.

85
00:04:22,279 --> 00:04:25,559
They might be maximum points, they might be minimum points,

86
00:04:25,560 --> 00:04:27,399
they might be inflection points, we don't know.

87
00:04:27,399 --> 00:04:29,849
They might be, you know, if this was a constant function,

88
00:04:29,850 --> 00:04:31,590
they could just be anything.

89
00:04:31,589 --> 00:04:34,689
So we really can't say a lot about them just yet, but they

90
00:04:34,689 --> 00:04:36,300
are points of interest.

91
00:04:36,300 --> 00:04:37,579
I guess that's all we can say.

92
00:04:37,579 --> 00:04:39,339
That they are definitely points of interest.

93
00:04:39,339 --> 00:04:41,449
But let's keep going, and let's try to understand the

94
00:04:41,449 --> 00:04:43,519
concativity, and maybe we can get a better sense

95
00:04:43,519 --> 00:04:45,259
of this graph.

96
00:04:45,259 --> 00:04:49,980
So let's figure out the second derivative.

97
00:04:49,980 --> 00:04:53,540
I'll do that in this orange color.

98
00:04:53,540 --> 00:04:59,319
So the second derivative of my function f, let's see, 3 times

99
00:04:59,319 --> 00:05:05,589
12 is 36x squared minus 24x.

100
00:05:05,589 --> 00:05:08,379


101
00:05:08,379 --> 00:05:09,219
So let's see.

102
00:05:09,220 --> 00:05:10,570
Well, there's a couple of things we can do.

103
00:05:10,569 --> 00:05:13,290
Now that we know the second derivative, we can answer the

104
00:05:13,290 --> 00:05:16,890
question, is my graph concave upwards or downwards at

105
00:05:16,889 --> 00:05:18,300
either of these points?

106
00:05:18,300 --> 00:05:20,270
So let's figure out what, at either of these

107
00:05:20,269 --> 00:05:21,779
critical points.

108
00:05:21,779 --> 00:05:23,009
And it'll all fit together.

109
00:05:23,009 --> 00:05:25,139
Remember, if it's concave upwards, then we're

110
00:05:25,139 --> 00:05:26,339
kind of in a U shape.

111
00:05:26,339 --> 00:05:29,369
If it's concave downwards, then we're in a kind of

112
00:05:29,370 --> 00:05:30,850
upside down U shape.

113
00:05:30,850 --> 00:05:34,400
So f prime prime, our second derivative, at x is equal

114
00:05:34,399 --> 00:05:36,799
to 0, is equal to what?

115
00:05:36,800 --> 00:05:40,530
It's equal to 36 0 squared minus 24 times 0.

116
00:05:40,529 --> 00:05:42,349
So that's just 0.

117
00:05:42,350 --> 00:05:45,210
So f prime prime is just equal to 0.

118
00:05:45,209 --> 00:05:49,120
We're neither concave upwards nor concave downwards here.

119
00:05:49,120 --> 00:05:50,420
It might be a transition point.

120
00:05:50,420 --> 00:05:50,990
It may not.

121
00:05:50,990 --> 00:05:52,850
If it is a transition point, then we're dealing with

122
00:05:52,850 --> 00:05:53,620
an inflection point.

123
00:05:53,620 --> 00:05:55,189
We're not sure yet.

124
00:05:55,189 --> 00:05:57,910
Now let's see what f prime prime, our second derivative,

125
00:05:57,910 --> 00:06:00,189
evaluated at 1 is.

126
00:06:00,189 --> 00:06:03,579
So that's 36 times 1, let me write it down, that's equal

127
00:06:03,579 --> 00:06:11,909
to 36 times 1 squared, which is 36, minus 24 times 1.

128
00:06:11,910 --> 00:06:16,650
So it's 36 minus 24, so it's equal to 12.

129
00:06:16,649 --> 00:06:20,149
So this is positive, our second derivative is positive here.

130
00:06:20,149 --> 00:06:22,879
It's equal to 12, which means our first derivative,

131
00:06:22,879 --> 00:06:24,500
our slope is increasing.

132
00:06:24,500 --> 00:06:26,949
The rate of change of our slope is positive here.

133
00:06:26,949 --> 00:06:30,944
So at this point right here, we are concave upwards.

134
00:06:30,944 --> 00:06:34,019


135
00:06:34,019 --> 00:06:36,199
Which tells me that this is probably a minimum

136
00:06:36,199 --> 00:06:36,769
point, right?

137
00:06:36,769 --> 00:06:40,549
The slope is 0 here, but we are concave upwards at that point.

138
00:06:40,550 --> 00:06:41,879
So that's interesting.

139
00:06:41,879 --> 00:06:43,920
So let's see if there any other potential

140
00:06:43,920 --> 00:06:44,780
inflection points here.

141
00:06:44,779 --> 00:06:49,459
We already know that this is a potential inflection point.

142
00:06:49,459 --> 00:06:50,680
Let me circle it in red.

143
00:06:50,680 --> 00:06:52,639
It's a potential inflection point.

144
00:06:52,639 --> 00:06:54,829
We don't know whether our function actually

145
00:06:54,829 --> 00:06:55,969
transitions at that point.

146
00:06:55,970 --> 00:06:58,310
We'll have to experiment a little bit to see if

147
00:06:58,310 --> 00:06:59,129
that's really the case.

148
00:06:59,129 --> 00:07:02,860
But let's see if there any other inflection points, or

149
00:07:02,860 --> 00:07:04,139
potential inflection points.

150
00:07:04,139 --> 00:07:07,050
So let's see if this equals 0 anywhere else.

151
00:07:07,050 --> 00:07:11,790
So 36 x squared minus 24 x is equal to 0.

152
00:07:11,790 --> 00:07:13,240
Let's solve for x.

153
00:07:13,240 --> 00:07:17,780
Let us factor out, well, we can factor out 12x.

154
00:07:17,779 --> 00:07:25,069
12x times 3x, right, 3x times 12x is 36x squared,

155
00:07:25,069 --> 00:07:28,560
minus 2, is equal to 0.

156
00:07:28,560 --> 00:07:31,089
So these two are equivalent expressions.

157
00:07:31,089 --> 00:07:33,519
If you multiply this out, you'll get this thing up here.

158
00:07:33,519 --> 00:07:36,509
So this thing is going to be equal to 0, either if 12 x is

159
00:07:36,509 --> 00:07:39,709
equal to 0, so 12 x is equal to zero, that gives us

160
00:07:39,709 --> 00:07:41,239
x is equal to zero.

161
00:07:41,240 --> 00:07:43,879
So at x equals 0, this thing equals 0.

162
00:07:43,879 --> 00:07:46,850
So the second derivative is 0 there, and we already knew

163
00:07:46,850 --> 00:07:48,689
that, because we tested that number out.

164
00:07:48,689 --> 00:07:52,180
Or this thing, if this expression was 0, then the

165
00:07:52,180 --> 00:07:54,970
entire second derivative would also be zero.

166
00:07:54,970 --> 00:07:55,740
So let's write that.

167
00:07:55,740 --> 00:08:03,090
So 3x minus 2 is equal to 0, 3x is equal to 2, just adding 2 to

168
00:08:03,089 --> 00:08:07,279
both sides, 3x is equal to 2/3.

169
00:08:07,279 --> 00:08:11,309
So this is another interesting point that we haven't really

170
00:08:11,310 --> 00:08:14,050
hit upon before that might be an inflection point.

171
00:08:14,050 --> 00:08:15,620
The reason why is it might be, is because the

172
00:08:15,620 --> 00:08:18,329
second derivative is definitely 0 here.

173
00:08:18,329 --> 00:08:20,579
You put 2/3 here, you're going to get 0.

174
00:08:20,579 --> 00:08:22,829
So what we have to do, is see whether the second derivative

175
00:08:22,829 --> 00:08:28,050
is positive or negative on either side of 2/3.

176
00:08:28,050 --> 00:08:29,199
We already have a sense of that.

177
00:08:29,199 --> 00:08:31,539
I mean, we could try out a couple of numbers.

178
00:08:31,540 --> 00:08:37,210
We know that, you know, if we say that x is greater than 2/3.

179
00:08:37,210 --> 00:08:38,690
Let me scroll down a little bit, just so

180
00:08:38,690 --> 00:08:40,420
we have some space.

181
00:08:40,419 --> 00:08:50,409
So let's see what happens when x is greater than 2/3,

182
00:08:50,409 --> 00:08:51,579
what is f prime prime?

183
00:08:51,580 --> 00:08:53,470
What is the second derivative?

184
00:08:53,470 --> 00:08:55,899
So let's try out a value that's pretty close, just

185
00:08:55,899 --> 00:08:59,019
to get a sense of things.

186
00:08:59,019 --> 00:09:04,189
So let me rewrite it. f prime prime of x is equal to,

187
00:09:04,190 --> 00:09:05,080
let me write like this.

188
00:09:05,080 --> 00:09:06,550
I mean, I could write like that, but this might be

189
00:09:06,549 --> 00:09:07,490
easier to deal with.

190
00:09:07,490 --> 00:09:13,029
It's equal to 12x times 3x minus 2.

191
00:09:13,029 --> 00:09:16,789
So if x is greater than 2/3, this term right here is

192
00:09:16,789 --> 00:09:18,269
going to be positive.

193
00:09:18,269 --> 00:09:21,240
That's definitely, any positive number times 12

194
00:09:21,240 --> 00:09:22,560
is going to be positive.

195
00:09:22,559 --> 00:09:25,389
But what about this term, right here?

196
00:09:25,389 --> 00:09:29,269
3 times 2/3 minus 2 is exactly 0, right?

197
00:09:29,269 --> 00:09:30,490
That's 2 minus 2.

198
00:09:30,490 --> 00:09:33,269
But anything larger than that, 3 times, you know, if I had

199
00:09:33,269 --> 00:09:38,669
2.1/3, this is going to be a positive quantity.

200
00:09:38,669 --> 00:09:41,669
Any value of x greater than 2/3 will make this thing

201
00:09:41,669 --> 00:09:43,579
right here positive.

202
00:09:43,580 --> 00:09:44,400
Right?

203
00:09:44,399 --> 00:09:46,669
This thing is also going to be positive.

204
00:09:46,669 --> 00:09:50,250
So that means that when x is greater than 2/3, that

205
00:09:50,250 --> 00:09:54,700
tells us that the second derivative is positive.

206
00:09:54,700 --> 00:09:56,800
It is greater than 0.

207
00:09:56,799 --> 00:10:00,529
So in our domain, as long as x is larger than 2/3,

208
00:10:00,529 --> 00:10:02,110
we are concave upwards.

209
00:10:02,110 --> 00:10:05,200
And we saw that here, at x is equal to 1.

210
00:10:05,200 --> 00:10:06,840
We were concave upwards.

211
00:10:06,840 --> 00:10:10,240
But what about x being less than 2/3?

212
00:10:10,240 --> 00:10:13,539
So when x is less than 2/3, let me write it, let me

213
00:10:13,539 --> 00:10:15,719
scroll down a little bit.

214
00:10:15,720 --> 00:10:19,170
When x is less than 2/3, what's going on?

215
00:10:19,169 --> 00:10:20,319
I'll rewrite it.

216
00:10:20,320 --> 00:10:22,500
f prime prime of x, second derivative,

217
00:10:22,500 --> 00:10:27,000
12x times 3x minus 2.

218
00:10:27,000 --> 00:10:30,019
Well, if we go really far left, we're going to get a negative

219
00:10:30,019 --> 00:10:31,230
number here, and this might be [UNINTELLIGIBLE].

220
00:10:31,230 --> 00:10:35,019
But if we just go right below 2/3, when we're still

221
00:10:35,019 --> 00:10:36,699
in the positive domain.

222
00:10:36,700 --> 00:10:41,250
So if this was like 1.9/3, which is a mix of a decimal and

223
00:10:41,250 --> 00:10:43,639
a fraction, or even 1/3, this thing is still going

224
00:10:43,639 --> 00:10:44,809
to be positive.

225
00:10:44,809 --> 00:10:47,909
Right below 2/3, this thing is still going to be positive.

226
00:10:47,909 --> 00:10:50,264
We're going to be multiplying 12 by a positive number.

227
00:10:50,264 --> 00:10:53,309
But what's going on right here?

228
00:10:53,309 --> 00:10:55,849


229
00:10:55,850 --> 00:10:57,759
At 2/3, we're exactly 0.

230
00:10:57,759 --> 00:11:00,700
But as you go to anything less than 2/3, 3

231
00:11:00,700 --> 00:11:02,230
times 1/3 is only 1.

232
00:11:02,230 --> 00:11:04,820
1 minus 2, you're going to get negative numbers.

233
00:11:04,820 --> 00:11:07,280
So when x is less than 2/3, this thing right here is

234
00:11:07,279 --> 00:11:09,289
going to be negative.

235
00:11:09,289 --> 00:11:13,480
So the second derivative, if x is less than 2/3, the second

236
00:11:13,480 --> 00:11:16,370
derivative, right to the left, right when you go less than

237
00:11:16,370 --> 00:11:20,269
2/3, the seconds derivative of x is less than 0.

238
00:11:20,269 --> 00:11:23,689
Now the fact that we have this transition, from when we're

239
00:11:23,690 --> 00:11:28,115
less than 2/3, we have a negative second derivative, and

240
00:11:28,115 --> 00:11:30,340
when we're greater than 2/3, we have a positive second

241
00:11:30,340 --> 00:11:34,300
derivative, that tells us that this, indeed, is an

242
00:11:34,299 --> 00:11:36,799
inflection point.

243
00:11:36,799 --> 00:11:40,240
That x is equal to 2/3 thirds is definitely an inflection

244
00:11:40,240 --> 00:11:43,940
point for our original function up here.

245
00:11:43,940 --> 00:11:47,180
Now, we have one more candidate inflection point, and then

246
00:11:47,179 --> 00:11:48,209
we're ready to graph.

247
00:11:48,210 --> 00:11:49,970
Then, you know, once you do all the inflection points and the

248
00:11:49,970 --> 00:11:53,700
max and the minimum, you are ready to graph the function.

249
00:11:53,700 --> 00:11:56,460
So let's see if x is equal to 0 is an inflection point.

250
00:11:56,460 --> 00:11:59,519
We know that the second derivative is 0 at 0.

251
00:11:59,519 --> 00:12:02,379
But what happens above and below the second derivative?

252
00:12:02,379 --> 00:12:04,669
So let me do our little test here.

253
00:12:04,669 --> 00:12:09,309
So when x is, let me draw a line so we don't get confused

254
00:12:09,309 --> 00:12:12,069
with all of the stuff that I wrote here.

255
00:12:12,070 --> 00:12:16,400
So when x is greater than 0, what's happening in

256
00:12:16,399 --> 00:12:17,840
the second derivative?

257
00:12:17,840 --> 00:12:21,850
Remember, the second derivative was equal to

258
00:12:21,850 --> 00:12:24,790
12x times 3x minus 2.

259
00:12:24,789 --> 00:12:27,009
I like writing it this way, because you've kind of

260
00:12:27,009 --> 00:12:29,830
decomposed it into two linear expressions, and you could see

261
00:12:29,830 --> 00:12:32,030
whether each of them are positive or negative.

262
00:12:32,029 --> 00:12:35,720
So if x is greater than 0, this thing right here is definitely

263
00:12:35,720 --> 00:12:39,899
going to be positive, and then this thing right here, right

264
00:12:39,899 --> 00:12:42,819
when you go right above x is greater than 0, so we have to

265
00:12:42,820 --> 00:12:45,810
make sure to be very close to this number, right?

266
00:12:45,809 --> 00:12:50,359
So this number, let's say it's 0.1.

267
00:12:50,360 --> 00:12:52,070
You're right above 0.

268
00:12:52,070 --> 00:12:54,260
So this isn't going to be true for all of x greater than 0.

269
00:12:54,259 --> 00:12:56,000
We just want to test exactly what happens, right when

270
00:12:56,000 --> 00:12:57,919
we go right above 0.

271
00:12:57,919 --> 00:12:59,019
So this is 0.1.

272
00:12:59,019 --> 00:13:02,079
You would have 0.3, 0.3 minus 2, that would be a

273
00:13:02,080 --> 00:13:03,520
negative number, right?

274
00:13:03,519 --> 00:13:06,470
So right as x goes right above 0, this thing

275
00:13:06,470 --> 00:13:08,410
right here is negative.

276
00:13:08,409 --> 00:13:12,629
So at x is greater than 0, you will have your second

277
00:13:12,629 --> 00:13:15,250
derivative is going to be less than 0.

278
00:13:15,250 --> 00:13:17,009
You're concave downwards.

279
00:13:17,009 --> 00:13:19,529
Which makes sense, because at some point, we're going to

280
00:13:19,529 --> 00:13:20,899
be hitting a transition.

281
00:13:20,899 --> 00:13:23,429
Remember, we were concave downwards before we

282
00:13:23,429 --> 00:13:25,250
got to 2/3, right?

283
00:13:25,250 --> 00:13:26,399
So this is consistent.

284
00:13:26,399 --> 00:13:30,579
From 0 to 2/3, we are concave downwards, and then at 2/3,

285
00:13:30,580 --> 00:13:32,540
we become concave upwards.

286
00:13:32,539 --> 00:13:38,389
Now let's see what happens when x is right less than, when x

287
00:13:38,389 --> 00:13:42,399
is just barely, just barely less than 0.

288
00:13:42,399 --> 00:13:45,519
So once again, f prime, the second derivitive of x is equal

289
00:13:45,519 --> 00:13:49,029
to 12x times 3x minus 2.

290
00:13:49,029 --> 00:13:49,589
Well, right.

291
00:13:49,590 --> 00:13:54,830
If x was minus 0.1 or 0.0001, no matter what, this thing is

292
00:13:54,830 --> 00:13:57,470
going to be negative, this expression right here is going

293
00:13:57,470 --> 00:14:01,410
to be negative, the 12x, right, you just have some negative

294
00:14:01,409 --> 00:14:03,909
value here, times 12, is going to be negative.

295
00:14:03,909 --> 00:14:05,939
And then what's this going to be?

296
00:14:05,940 --> 00:14:10,000
Well, 3 times minus 0.1 is going to be minus 0.3,

297
00:14:10,000 --> 00:14:12,090
minus 2 is minus 2.3.

298
00:14:12,090 --> 00:14:13,250
You're definitely going to have a negative.

299
00:14:13,250 --> 00:14:15,363
This value right here is going to be negative, and then when

300
00:14:15,363 --> 00:14:17,075
you subtract from a negative, it's definitely going

301
00:14:17,075 --> 00:14:18,120
to be negative.

302
00:14:18,120 --> 00:14:20,269
So that is also going to be negative.

303
00:14:20,269 --> 00:14:22,090
But if you multiply a negative times a negative, you're

304
00:14:22,090 --> 00:14:23,290
going to get a positive.

305
00:14:23,289 --> 00:14:27,659
So actually, right below x is less than 0, the second

306
00:14:27,659 --> 00:14:28,454
derivative is positive.

307
00:14:28,455 --> 00:14:32,420


308
00:14:32,419 --> 00:14:35,139
Now, this all might have been a little bit confusing, but we

309
00:14:35,139 --> 00:14:37,120
should now have the payoff.

310
00:14:37,120 --> 00:14:38,389
We now have the payoff.

311
00:14:38,389 --> 00:14:41,049
We have all of the interesting things going on.

312
00:14:41,049 --> 00:14:46,059
We know that at x is equal to 1, we know that at x is equal

313
00:14:46,059 --> 00:14:48,689
to 1, let me write it over here.

314
00:14:48,690 --> 00:14:55,106
We've figured out at x is equal to 1, the slope is 0.

315
00:14:55,105 --> 00:14:59,840
So f prime prime is, sorry, let me write this way.

316
00:14:59,840 --> 00:15:03,500
I should have said, we know that the slope is 0.

317
00:15:03,500 --> 00:15:04,669
Slope is equal to 0.

318
00:15:04,669 --> 00:15:07,019
And we figured that out because the first derivative was 0.

319
00:15:07,019 --> 00:15:09,000
This was a critical point.

320
00:15:09,000 --> 00:15:12,860
And we know that we're dealing with, the function is concave

321
00:15:12,860 --> 00:15:14,860
upwards at this point.

322
00:15:14,860 --> 00:15:17,740
And that tells us that this is going to be a minimum point.

323
00:15:17,740 --> 00:15:21,100


324
00:15:21,100 --> 00:15:22,519
And we should actually get the coordinates so we

325
00:15:22,519 --> 00:15:23,389
can actually graph it.

326
00:15:23,389 --> 00:15:24,980
That was the whole point of this video.

327
00:15:24,980 --> 00:15:29,600
So f of 1 is equal to what?

328
00:15:29,600 --> 00:15:33,509
f of 1, let's go back to our original function, is 3 times

329
00:15:33,509 --> 00:15:38,189
1, right, 1 to the fourth is just 1, 3 times 1 minus

330
00:15:38,190 --> 00:15:40,810
4 plus 2, right?

331
00:15:40,809 --> 00:15:47,509
So it's 3 times 1 minus 4 times 1, which is minus 1, plus 2,

332
00:15:47,509 --> 00:15:48,929
well, that's just a positive 1.

333
00:15:48,929 --> 00:15:50,919
So f of 1 is one.

334
00:15:50,919 --> 00:15:56,419
And then we know at x is equal to 0, we also figured out that

335
00:15:56,419 --> 00:15:59,259
the slope is equal to 0.

336
00:15:59,259 --> 00:16:02,019
But we figured out that this was an inflection point, right?

337
00:16:02,019 --> 00:16:05,039
The concativity switches before and after.

338
00:16:05,039 --> 00:16:06,509
So this is an inflection point.

339
00:16:06,509 --> 00:16:10,080


340
00:16:10,080 --> 00:16:15,700
And we are concave below 0, so when x is less

341
00:16:15,700 --> 00:16:17,535
than 0, we are upwards.

342
00:16:17,534 --> 00:16:20,039


343
00:16:20,039 --> 00:16:22,179
Our second derivative is positive.

344
00:16:22,179 --> 00:16:25,250
And when x is it greater than 0, we are downwards.

345
00:16:25,250 --> 00:16:26,419
We're concave downwards.

346
00:16:26,419 --> 00:16:28,399
Right above, not for all of the [? domain ?]

347
00:16:28,399 --> 00:16:32,759
x and 0, just right above 0, downwards.

348
00:16:32,759 --> 00:16:35,519
And then what is f of 0, just so we know, because we

349
00:16:35,519 --> 00:16:37,659
want to graph that point?

350
00:16:37,659 --> 00:16:39,959
f of 0.

351
00:16:39,960 --> 00:16:41,620
See, f of 0, this is easy.

352
00:16:41,620 --> 00:16:45,379
3 times 0 minus 4 times 0 plus 2, that's just 2.

353
00:16:45,379 --> 00:16:48,080
f of 0 is 2.

354
00:16:48,080 --> 00:16:50,629
And then finally we got the point, x is equal to 2/3.

355
00:16:50,629 --> 00:16:53,269


356
00:16:53,269 --> 00:16:56,419
Let me do that in another color.

357
00:16:56,419 --> 00:17:00,969
We had the point x is equal to 2/3.

358
00:17:00,970 --> 00:17:05,670
We figured out that this was an inflection point.

359
00:17:05,670 --> 00:17:09,670
The slope definitely isn't 0 there, because it wasn't one

360
00:17:09,670 --> 00:17:11,740
of the critical points.

361
00:17:11,740 --> 00:17:15,390
And we know that we are downwards.

362
00:17:15,390 --> 00:17:19,210
We know that when x is less than 2/3, or right less than

363
00:17:19,210 --> 00:17:22,100
2/3, we are concave downwards.

364
00:17:22,099 --> 00:17:27,119
And when x is greater than 2/3, we saw it up here, when x was

365
00:17:27,119 --> 00:17:31,809
greater than 2/3, right up here, we were concave upwards.

366
00:17:31,809 --> 00:17:35,039
The second derivative was positive.

367
00:17:35,039 --> 00:17:35,884
We were upwards.

368
00:17:35,884 --> 00:17:38,460


369
00:17:38,460 --> 00:17:40,789
Now we could actually figure out, what's f of 2/3?

370
00:17:40,789 --> 00:17:43,420
That's actually a little bit complicated.

371
00:17:43,420 --> 00:17:44,529
We don't even have to figure that out, I

372
00:17:44,529 --> 00:17:45,319
don't think, to graph.

373
00:17:45,319 --> 00:17:48,179
I think we could do a pretty good job of graphing it just

374
00:17:48,180 --> 00:17:49,269
with what we know right now.

375
00:17:49,269 --> 00:17:50,369
So that's our take aways.

376
00:17:50,369 --> 00:17:52,739
Let me do a rough graph.

377
00:17:52,740 --> 00:17:54,339
Let's see.

378
00:17:54,339 --> 00:17:57,839
So let me do my axes, just like that.

379
00:17:57,839 --> 00:18:02,199
So we're going to want to graph the point 0, 2.

380
00:18:02,200 --> 00:18:04,120
So let's say that the point, 0, 2.

381
00:18:04,119 --> 00:18:08,569
So this is x is equal to 0, and we go up, 1, 2.

382
00:18:08,569 --> 00:18:10,609
So this is the point 0, 2.

383
00:18:10,609 --> 00:18:12,639
Maybe I'll do it in that color, the color I was

384
00:18:12,640 --> 00:18:14,480
using, so that's this color.

385
00:18:14,480 --> 00:18:17,049
So that's that point right there.

386
00:18:17,049 --> 00:18:19,649
Then we have the point x, we have f of 1, which is

387
00:18:19,650 --> 00:18:21,930
the point 1, 1, right?

388
00:18:21,930 --> 00:18:23,090
So this point right here.

389
00:18:23,089 --> 00:18:26,099


390
00:18:26,099 --> 00:18:29,149
So that's the point 1, 1.

391
00:18:29,150 --> 00:18:32,250
This was the point 0, 2.

392
00:18:32,250 --> 00:18:36,259
And then we have x is equal to 2/3, which is our

393
00:18:36,259 --> 00:18:36,940
inflection point.

394
00:18:36,940 --> 00:18:39,730
So when x is 2/3, we don't know exactly what

395
00:18:39,730 --> 00:18:41,110
number f of 2/3 is.

396
00:18:41,109 --> 00:18:42,349
Maybe here someplace.

397
00:18:42,349 --> 00:18:43,889
Let's say f of 2/3 is right there.

398
00:18:43,890 --> 00:18:48,840
So that's the point, 2/3, and then wherever f of 2/3 is.

399
00:18:48,839 --> 00:18:50,959
It looks like it's going to be 1 point something.

400
00:18:50,960 --> 00:18:52,279
f of 2/3.

401
00:18:52,279 --> 00:18:53,839
You could calculate it, if you like, you just have

402
00:18:53,839 --> 00:18:55,500
to substitute back in the function.

403
00:18:55,500 --> 00:18:59,440
But we're ready to graph this thing.

404
00:18:59,440 --> 00:19:04,100
So we know that at x is equal to 1, the slope is 0.

405
00:19:04,099 --> 00:19:05,109
We know that the slope is 0.

406
00:19:05,109 --> 00:19:05,750
It's flat here.

407
00:19:05,750 --> 00:19:07,519
We know it's concave upwards.

408
00:19:07,519 --> 00:19:12,069
So we're dealing, it looks like this, looks like

409
00:19:12,069 --> 00:19:13,029
that over that interval.

410
00:19:13,029 --> 00:19:14,480
We're concave upwards.

411
00:19:14,480 --> 00:19:17,440
And we know we're concave upwards from x is equal

412
00:19:17,440 --> 00:19:19,480
to 2/3 and on, right?

413
00:19:19,480 --> 00:19:20,690
Let me do it in that color.

414
00:19:20,690 --> 00:19:23,309
We knew x equals 2/3 and on, we're concave upwards.

415
00:19:23,309 --> 00:19:25,669
And that's why I was able to draw this U-shape.

416
00:19:25,670 --> 00:19:29,390
Now we know that when x is less than 2/3 and greater than

417
00:19:29,390 --> 00:19:31,759
0, we're concave downwards.

418
00:19:31,759 --> 00:19:33,740
So the graph would look something like this,

419
00:19:33,740 --> 00:19:34,269
over this interval.

420
00:19:34,269 --> 00:19:36,889
We'll be concave downwards.

421
00:19:36,890 --> 00:19:38,820
Let me draw it nicely.

422
00:19:38,819 --> 00:19:41,079
Over this interval, the slope is decreasing.

423
00:19:41,079 --> 00:19:42,949
And you could see it, if you keep drawing tangent lines.

424
00:19:42,950 --> 00:19:45,660
It's flattish there, it gets negative, more negative, more

425
00:19:45,660 --> 00:19:48,490
negative, until the inflection point, and then it starts

426
00:19:48,490 --> 00:19:51,329
increasing again, because we go back to concave upwards.

427
00:19:51,329 --> 00:19:54,599
And then finally, the last interval is below 0, and we

428
00:19:54,599 --> 00:19:58,279
know below 0, when x is less than 0, we're concave upwards.

429
00:19:58,279 --> 00:20:01,109
So the graph looks like this.

430
00:20:01,109 --> 00:20:02,819
The graph looks like that.

431
00:20:02,819 --> 00:20:05,490
And we also know that x is equal to 0 was a critical

432
00:20:05,490 --> 00:20:06,620
point, the slope of 0.

433
00:20:06,619 --> 00:20:08,819
So the graph is actually flat right there, too.

434
00:20:08,819 --> 00:20:11,879
So this is an inflection point where the slope was also 0.

435
00:20:11,880 --> 00:20:13,260
So this is our final graph.

436
00:20:13,259 --> 00:20:14,059
We're done.

437
00:20:14,059 --> 00:20:16,549
After all that work, we were able to use our calculus

438
00:20:16,549 --> 00:20:18,589
skills, and our knowledge of inflection points, and

439
00:20:18,589 --> 00:20:21,599
concativity, and transitions in concativity, to

440
00:20:21,599 --> 00:20:25,509
actually graph this fairly hairy-looking graph.

441
00:20:25,509 --> 00:20:28,394
But this should be kind of what it looks like, if you graph

442
00:20:28,394 --> 00:20:30,099
it on your calculator.

443
00:20:30,099 --> 00:20:30,865