1
00:00:00,256 --> 00:00:03,652
- [Voiceover] The figure above
shows the graph of f prime,

2
00:00:03,652 --> 00:00:06,612
the derivative of a
twice-differentiable function f,

3
00:00:06,612 --> 00:00:08,047
on the interval, that's a closed interval,

4
00:00:08,047 --> 00:00:09,580
from negative three to four.

5
00:00:09,580 --> 00:00:12,262
The graph of f prime
has horizontal tangents

6
00:00:12,262 --> 00:00:14,526
at x equals negative one, x equals one,

7
00:00:14,526 --> 00:00:15,860
and x equals three.

8
00:00:15,860 --> 00:00:19,192
So you have a horizontal
tangent right over,

9
00:00:19,192 --> 00:00:21,921
a horizontal tangent right over there.

10
00:00:23,041 --> 00:00:24,524
And let me draw that a little bit neater,

11
00:00:24,524 --> 00:00:25,458
right over there,

12
00:00:25,458 --> 00:00:27,320
a horizontal tangent right over there,

13
00:00:27,320 --> 00:00:29,362
and a horizontal tangent right over there.

14
00:00:30,976 --> 00:00:31,557
Alright.

15
00:00:31,557 --> 00:00:33,345
The areas of the regions bounded

16
00:00:33,345 --> 00:00:36,475
by the x-axis and the graph of f prime

17
00:00:36,475 --> 00:00:38,731
on the intervals negative two to one,

18
00:00:38,731 --> 00:00:40,346
closed intervals from negative two to one,

19
00:00:40,346 --> 00:00:42,694
so this region right over here,

20
00:00:42,694 --> 00:00:45,142
and the region from one to four,

21
00:00:45,142 --> 00:00:46,591
so this region right over there,

22
00:00:46,591 --> 00:00:48,494
they tell us have,

23
00:00:48,494 --> 00:00:50,911
have the areas are 9 and 12, respectively.

24
00:00:50,911 --> 00:00:54,203
So that area's 9 and that area is 12.

25
00:00:54,203 --> 00:00:55,729
So Part A,

26
00:00:55,729 --> 00:00:59,955
Find all x coordinates at
which f has a relative maximum.

27
00:00:59,955 --> 00:01:03,473
Give a reason for your answer.

28
00:01:03,473 --> 00:01:05,527
All x-coordinates at which f

29
00:01:05,527 --> 00:01:06,955
has a relative maximum.

30
00:01:06,955 --> 00:01:07,629
So you might say,

31
00:01:07,629 --> 00:01:09,196
"Oh wait, wait this looks
like a relative maximum

32
00:01:09,196 --> 00:01:10,509
over here, but this isn't f.

33
00:01:10,509 --> 00:01:12,413
This is the graph of f prime."

34
00:01:12,413 --> 00:01:13,793
So let's think about when we used to

35
00:01:13,793 --> 00:01:15,490
we don't have a graph of f in front of us.

36
00:01:15,490 --> 00:01:17,427
So let's think about
what it needs to be true

37
00:01:17,427 --> 00:01:20,943
for f to have a relative
maximum at a point.

38
00:01:20,943 --> 00:01:23,210
We are probably familiar with

39
00:01:23,210 --> 00:01:25,415
what relative maximum will look like.

40
00:01:25,415 --> 00:01:28,214
It'd look like a little lump, like that.

41
00:01:28,214 --> 00:01:30,072
It could also actually look like that

42
00:01:30,072 --> 00:01:32,173
but since this is differentiable function

43
00:01:32,173 --> 00:01:33,671
over the interval, we're
probably not dealing

44
00:01:33,671 --> 00:01:36,213
with a relative maximum
that looks like that.

45
00:01:37,571 --> 00:01:41,803
And so what do we know about
a relative maximum point?

46
00:01:41,803 --> 00:01:44,754
So let's say that's our relative maximum.

47
00:01:44,754 --> 00:01:46,883
As we approach our relative maximum

48
00:01:46,883 --> 00:01:50,202
from values, as we have x
values that are approaching

49
00:01:50,202 --> 00:01:54,043
the x value of our relative maximum point,

50
00:01:54,043 --> 00:01:56,659
as we approach it from
values below that x value,

51
00:01:56,659 --> 00:01:59,699
we see that we have a positive slope.

52
00:01:59,699 --> 00:02:02,839
Our function needs to be increasing.

53
00:02:02,839 --> 00:02:04,751
So over here.

54
00:02:05,784 --> 00:02:08,686
Over here we see f is increasing,

55
00:02:08,686 --> 00:02:11,827
going into the relative maximum point,

56
00:02:11,827 --> 00:02:12,726
f is increasing,

57
00:02:12,726 --> 00:02:15,227
which means that the derivative of f

58
00:02:15,227 --> 00:02:17,312
the derivative of f must
be greater than zero.

59
00:02:18,342 --> 00:02:20,874
And then after we past that maximum point,

60
00:02:20,874 --> 00:02:22,491
after we past that maximum point,

61
00:02:22,491 --> 00:02:26,247
we see that our function
needs to be decreasing.

62
00:02:26,247 --> 00:02:27,569
Do this in another color.

63
00:02:27,569 --> 00:02:29,711
We see that our funciton is decreasing

64
00:02:29,711 --> 00:02:33,090
right over here, so f decreasing,

65
00:02:34,100 --> 00:02:36,764
decreasing, which means

66
00:02:36,764 --> 00:02:39,684
that f prime of x needs
to be less than zero.

67
00:02:40,524 --> 00:02:43,745
So our relative maximum point

68
00:02:43,745 --> 00:02:46,247
should happen at an x value.

69
00:02:46,247 --> 00:02:47,312
It should happen at an x value

70
00:02:47,312 --> 00:02:50,284
where our first derivative transitions

71
00:02:50,284 --> 00:02:52,478
from being greater than zero

72
00:02:52,478 --> 00:02:55,244
to being less than zero.

73
00:02:55,244 --> 00:02:56,437
So what x value,

74
00:02:56,437 --> 00:02:57,262
so let me say this,

75
00:02:57,262 --> 00:03:00,737
so we have

76
00:03:00,737 --> 00:03:05,407
f has relative, let me
just write it shorthand,

77
00:03:05,407 --> 00:03:10,351
relative maximum at x values

78
00:03:11,505 --> 00:03:13,526
where

79
00:03:13,526 --> 00:03:16,561
f prime transitions,

80
00:03:16,561 --> 00:03:17,631
transitions,

81
00:03:19,022 --> 00:03:21,703
transitions from

82
00:03:21,703 --> 00:03:26,272
from positive, positive,

83
00:03:27,871 --> 00:03:32,713
to negative, to, let me write
this a little bit neater,

84
00:03:32,713 --> 00:03:34,493
to negative.

85
00:03:35,653 --> 00:03:37,713
to negative.

86
00:03:37,713 --> 00:03:39,923
And where do we see f prime transitioning

87
00:03:39,923 --> 00:03:41,902
from positive to negative?

88
00:03:41,902 --> 00:03:43,933
Well over here we only
see that happening once.

89
00:03:43,933 --> 00:03:47,138
We see right here f prime is
positive, positive, positive,

90
00:03:47,138 --> 00:03:49,077
and then it goes negative,
negative, negative.

91
00:03:49,077 --> 00:03:50,797
So we see,

92
00:03:50,797 --> 00:03:54,143
we see f prime is positive over here,

93
00:03:55,273 --> 00:03:58,020
And then, right when we
hit x equals negative two,

94
00:03:58,020 --> 00:04:00,702
f prime becomes negative.

95
00:04:00,702 --> 00:04:03,849
F prime becomes negative.

96
00:04:03,849 --> 00:04:05,706
So we know that the function itself,

97
00:04:05,706 --> 00:04:06,426
not f prime,

98
00:04:06,426 --> 00:04:07,877
f must be increasing here

99
00:04:07,877 --> 00:04:10,768
because f prime is positive,

100
00:04:10,768 --> 00:04:13,369
and then our function at f

101
00:04:13,369 --> 00:04:14,518
is decreasing here

102
00:04:14,518 --> 00:04:17,084
because f prime is negative.

103
00:04:18,054 --> 00:04:20,969
And so this happens at x equals two.

104
00:04:20,969 --> 00:04:22,773
So let me write that down.

105
00:04:22,773 --> 00:04:24,870
This happens at x equals two.

106
00:04:24,870 --> 00:04:27,550
This happens,

107
00:04:27,550 --> 00:04:31,969
happens at x equals two.

108
00:04:31,969 --> 00:04:33,301
And we're done.