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HERBERT GROSS: Hi, our lecture
today, if we're looking at
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this from an analytical point
view, should be called 'Maxima
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00:00:38,140 --> 00:00:39,470
and Minima'.
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00:00:39,470 --> 00:00:41,510
And if we're looking at it
from a geometric point of
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view, 'High Points
and Low Points'.
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And actually, whichever point of
view we're looking at from,
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00:00:47,530 --> 00:00:51,400
it's a very nice application off
much of the theory that we
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00:00:51,400 --> 00:00:53,260
have learned up until now.
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00:00:53,260 --> 00:00:56,330
So without further ado, let's
talk a little bit about what
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00:00:56,330 --> 00:01:00,830
we mean by high points, low
points, maxima or minima.
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Which is, as I say, I've called
the lecture today.
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Now, as is always the case, we
usually have to have some sort
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of a fundamental result
from which all of our
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00:01:09,480 --> 00:01:11,340
other results follow.
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00:01:11,340 --> 00:01:14,620
And in this context what I call
the fundamental theorem
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00:01:14,620 --> 00:01:18,020
for a study of maxima minima
is the following.
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00:01:18,020 --> 00:01:23,765
Suppose that 'f of c' is at
least as great as 'f of x' for
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00:01:23,765 --> 00:01:26,540
all 'x' in a delta neighborhood
of 'c'.
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00:01:26,540 --> 00:01:31,090
In other words, we have some
open interval with delta
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00:01:31,090 --> 00:01:32,570
surrounding 'c'.
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00:01:32,570 --> 00:01:36,290
And then for all 'x' in that
neighborhood, 'f of c' is at
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00:01:36,290 --> 00:01:38,110
least as great as 'f of x'.
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00:01:38,110 --> 00:01:43,690
Or equivalently, it might be
that 'f of c' is less than or
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00:01:43,690 --> 00:01:47,270
equal to 'f of x' for all 'x'
in this neighborhood.
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00:01:47,270 --> 00:01:51,280
And suppose also, that 'f
prime of c' exists.
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00:01:51,280 --> 00:01:53,900
Then the theorem says, 'f
prime of c' in this
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00:01:53,900 --> 00:01:56,310
case, must be 0.
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00:01:56,310 --> 00:01:59,530
And whereas this can be proven
analytically, again the
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00:01:59,530 --> 00:02:02,190
analytical proof is motivated
by what's happening
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00:02:02,190 --> 00:02:03,160
geometrically.
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00:02:03,160 --> 00:02:05,700
And the geometric demonstration
is particularly
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00:02:05,700 --> 00:02:08,380
easy to visualize in this
particular case.
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Let me run the risk of drawing
this fairly freehand here.
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See what we're saying is this.
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Suppose you have at the point
(c, 'f of c') on this
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00:02:23,470 --> 00:02:24,840
particular curve.
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00:02:24,840 --> 00:02:28,240
Suppose that this particular
curve the derivative exists
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00:02:28,240 --> 00:02:29,060
and is positive.
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00:02:29,060 --> 00:02:30,070
See that's the way
I've drawn this.
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00:02:30,070 --> 00:02:32,490
In other words, notice that in
the neighborhood of the point
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00:02:32,490 --> 00:02:36,200
'c' here, for example, the
derivative here is positive.
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00:02:36,200 --> 00:02:38,450
The curve is always rising.
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00:02:38,450 --> 00:02:41,540
Now, what we're saying over here
is that if you look at
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this particular picture, observe
that 'f of c' will not
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00:02:45,660 --> 00:02:48,700
exceed 'f of x' for all 'x'
in this neighborhood.
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00:02:48,700 --> 00:02:50,710
In fact, I think you can see
just by looking at this
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00:02:50,710 --> 00:02:55,920
picture that as 'x' increases,
'f of x' increases.
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00:02:55,920 --> 00:02:59,490
In other words, in terms of this
picture, if 'x' is less
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00:02:59,490 --> 00:03:04,140
than 'c', 'f of x' is
less than 'f of c'.
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00:03:04,140 --> 00:03:09,030
And if 'x' is greater than
'c', 'f of x' is
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00:03:09,030 --> 00:03:11,730
greater than 'f of c'.
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00:03:11,730 --> 00:03:14,750
In other words, what we're
saying is if the derivative is
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00:03:14,750 --> 00:03:17,800
positive, it means that the
curve is always rising.
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00:03:17,800 --> 00:03:20,840
And hence, the point in the
middle of that neighborhood
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cannot be the highest point
in that neighborhood.
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00:03:24,060 --> 00:03:26,350
Nor, for example, can it
be the lowest point.
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00:03:26,350 --> 00:03:30,910
In fact, a similar argument
holds if we want to illustrate
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00:03:30,910 --> 00:03:33,950
that 'f of c' is less than or
equal to 'f of x' in this
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00:03:33,950 --> 00:03:35,640
particular neighborhood.
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00:03:35,640 --> 00:03:39,470
And by the way, again, observe
what we're saying.
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00:03:39,470 --> 00:03:41,450
This is rather crucial
over here.
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00:03:41,450 --> 00:03:44,660
First of all, we're talking
about a sufficiently small
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00:03:44,660 --> 00:03:46,040
neighborhood of 'c'.
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00:03:46,040 --> 00:03:50,360
What I mean by that is
something like this.
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00:03:50,360 --> 00:03:52,910
And by the way, this could
cause a little bit of a
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problem if you're not careful
with your language.
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00:03:55,935 --> 00:03:58,930
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00:03:58,930 --> 00:04:02,120
If we drew a picture like this
and you say to somebody, where
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00:04:02,120 --> 00:04:03,740
are the high-low
points on this?
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00:04:03,740 --> 00:04:06,080
I think it's quite natural that
the person would say,
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well, this is the high point.
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00:04:07,820 --> 00:04:09,350
I'll call that 'H'.
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00:04:09,350 --> 00:04:10,630
And this is the low point.
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00:04:10,630 --> 00:04:12,270
I'll call that 'L'.
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And he might tend to forget
about this point here because
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00:04:16,180 --> 00:04:19,360
even though it's fairly high,
it's not nearly as high as
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00:04:19,360 --> 00:04:20,160
this point.
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00:04:20,160 --> 00:04:26,960
What I'd like you to see however
is that our definition
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00:04:26,960 --> 00:04:28,750
talks about what?
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00:04:28,750 --> 00:04:31,430
In a neighborhood of
the given point.
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00:04:31,430 --> 00:04:34,050
You see what we're saying here
is that if we pick a
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00:04:34,050 --> 00:04:37,470
particular neighborhood,
an appropriately chosen
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00:04:37,470 --> 00:04:41,190
neighborhood surrounding 'c',
then what is true is that at
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00:04:41,190 --> 00:04:45,760
the value of 'x' corresponding
to 'c', 'f of c' is the
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00:04:45,760 --> 00:04:49,900
highest point in a suitable
neighborhood of 'c'.
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00:04:49,900 --> 00:04:53,140
In other words, if you knew that
for some reason or other,
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00:04:53,140 --> 00:04:57,210
you had to be working in this
neighborhood here, you could
97
00:04:57,210 --> 00:05:00,980
say, well, in the domain in
which I'm interested in, this
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00:05:00,980 --> 00:05:02,730
is a high point.
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00:05:02,730 --> 00:05:07,170
And this is why we talk about
'local' or 'relative' maxima
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00:05:07,170 --> 00:05:13,080
or minima in addition to
'absolute' maxima and minima.
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00:05:13,080 --> 00:05:15,980
In fact, you see, this happens
quite frequently in practice.
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00:05:15,980 --> 00:05:18,420
Suppose you were doing an
experiment and you really
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00:05:18,420 --> 00:05:21,500
wanted to produce the largest
possible value of 'y'.
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00:05:21,500 --> 00:05:25,000
Well, you see, utopianaly you
would like to pick 'x' out
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00:05:25,000 --> 00:05:26,140
here someplace.
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00:05:26,140 --> 00:05:29,440
But suppose because of some
constraint in the laboratory,
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00:05:29,440 --> 00:05:31,930
the largest value of 'x'
that you could choose
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00:05:31,930 --> 00:05:33,430
might be over here.
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00:05:33,430 --> 00:05:35,670
And then you see what the
equivalent problem would be.
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00:05:35,670 --> 00:05:38,410
And this is where the domain
of the function has such a
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00:05:38,410 --> 00:05:42,520
powerful meaning in terms of
practical applications.
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00:05:42,520 --> 00:05:45,520
What you're saying is look-it ,
if the domain of my function
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00:05:45,520 --> 00:05:48,110
is limited to this interval
over here, then this
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00:05:48,110 --> 00:05:50,300
particular point as far
as I'm concerned
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00:05:50,300 --> 00:05:52,840
is the highest point.
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00:05:52,840 --> 00:05:56,265
In other words then, notice
that the labelling 'N sub
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00:05:56,265 --> 00:05:59,950
'delta of c'' indicates that
you're talking locally rather
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00:05:59,950 --> 00:06:03,640
than globally in a neighborhood
of 'c'.
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00:06:03,640 --> 00:06:06,140
And this is the important
issue here.
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00:06:06,140 --> 00:06:10,560
Now, the hardest part about this
particular result as I
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00:06:10,560 --> 00:06:15,520
see it, is not understanding the
result as much as it is of
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00:06:15,520 --> 00:06:18,570
reading more into the result
than what's really there.
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00:06:18,570 --> 00:06:21,420
And so I have a few
cautions for you.
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00:06:21,420 --> 00:06:23,800
The three commandments
they turn out to be.
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00:06:23,800 --> 00:06:27,510
The first is, beware
of false converses.
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00:06:27,510 --> 00:06:30,550
And before you can be beware of
false converses, you have
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00:06:30,550 --> 00:06:32,210
to know what a converse is.
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00:06:32,210 --> 00:06:36,790
Roughly speaking, a converse
applies only to an if-then
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00:06:36,790 --> 00:06:38,350
type of statement.
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00:06:38,350 --> 00:06:41,120
And you get the converse
of a given statement by
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00:06:41,120 --> 00:06:45,540
interchanging the clauses that
follow the if and the then.
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00:06:45,540 --> 00:06:47,860
And the reason you have to
beware is that a true
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00:06:47,860 --> 00:06:50,370
statement can have
a false converse.
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00:06:50,370 --> 00:06:53,250
For example, consider the
following true statement.
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00:06:53,250 --> 00:06:58,390
If a person is listening to me
lecture now, then he is alive.
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00:06:58,390 --> 00:07:01,140
Hopefully, a true statement.
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00:07:01,140 --> 00:07:04,590
If we now interchange the
clauses to form the converse
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00:07:04,590 --> 00:07:07,460
it says if a person is
alive, then he is
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00:07:07,460 --> 00:07:09,650
listening to me lecture.
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00:07:09,650 --> 00:07:12,420
A tragically false statement.
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00:07:12,420 --> 00:07:15,770
But notice the difference
between inverting the clauses
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00:07:15,770 --> 00:07:17,590
of an if-then statement.
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00:07:17,590 --> 00:07:18,870
And the idea is this.
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00:07:18,870 --> 00:07:25,010
Notice that our theorem says if
'f prime' exists, then-- or
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00:07:25,010 --> 00:07:28,270
if there is a local maximum
or a local minimum, 'f
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00:07:28,270 --> 00:07:29,960
prime of c' is 0.
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00:07:29,960 --> 00:07:34,020
It does not say that if 'f prime
of c' is 0 we have a
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00:07:34,020 --> 00:07:36,220
local maximum or a
local minimum.
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00:07:36,220 --> 00:07:38,560
Perhaps the easiest
way to see that is
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00:07:38,560 --> 00:07:40,720
in terms of an example.
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00:07:40,720 --> 00:07:43,790
That's the nicest thing to show
how a converse is false.
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00:07:43,790 --> 00:07:46,550
To prove that something is true,
you can't do it just by
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00:07:46,550 --> 00:07:48,350
showing its true in
certain cases.
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00:07:48,350 --> 00:07:51,180
But to show that something is
not true, all you have to do
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00:07:51,180 --> 00:07:54,060
is exhibit one example in which
the result is false.
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00:07:54,060 --> 00:07:56,780
That's enough to prove that
it can't always be true.
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00:07:56,780 --> 00:07:59,660
For example, in this particular
diagram, notice
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00:07:59,660 --> 00:08:02,340
that in the curve 'y' equals
'f of x', which I've drawn
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00:08:02,340 --> 00:08:06,900
here, the curve at the point
'c' comma 'f of c' has a
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00:08:06,900 --> 00:08:08,230
horizontal tangent.
161
00:08:08,230 --> 00:08:10,860
In other words, 'f prime
of c' is 0 here.
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00:08:10,860 --> 00:08:14,520
But notice that in any
neighborhood that surrounds
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00:08:14,520 --> 00:08:18,780
'c', in any open interval that's
around 'c', notice that
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00:08:18,780 --> 00:08:22,650
if 'x' is less than 'c', 'f of
x' will be less than 'f of c'.
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00:08:22,650 --> 00:08:25,610
And if 'x' is greater than 'c',
'f of x' will be greater
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00:08:25,610 --> 00:08:26,710
than 'f of c'.
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00:08:26,710 --> 00:08:29,000
In other words, notice that
except for the stationary
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00:08:29,000 --> 00:08:32,370
value at which we have a
horizontal tangent, the graph
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00:08:32,370 --> 00:08:36,460
is always rising in any
neighborhood of 'c'.
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00:08:36,460 --> 00:08:39,409
That's the first point
I want you to see.
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00:08:39,409 --> 00:08:42,860
The second point says
beware if 'f prime
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00:08:42,860 --> 00:08:44,670
of c' doesn't exist.
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00:08:44,670 --> 00:08:46,650
See all our theorem
said is, look it.
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00:08:46,650 --> 00:08:49,720
If you have a relative high
point, a relative low point,
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00:08:49,720 --> 00:08:53,900
relative max, or a relative min
at 'x' equal 'c', and if
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00:08:53,900 --> 00:09:00,050
'f prime of c' exists, then
'f prime of c' is 0.
177
00:09:00,050 --> 00:09:03,060
But who said that 'f prime
of c' has to exist?
178
00:09:03,060 --> 00:09:05,440
Again, let's look in terms
of an example.
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00:09:05,440 --> 00:09:09,830
If we let 'f of x' equal the
absolute value of 'x', recall
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00:09:09,830 --> 00:09:15,160
from our previous assignments
and the like that the
181
00:09:15,160 --> 00:09:18,050
derivative of 'f of x',
in this case, does not
182
00:09:18,050 --> 00:09:19,570
exist when 'x' is 0.
183
00:09:19,570 --> 00:09:26,500
In other words, 'f prime
of 0' doesn't exist.
184
00:09:26,500 --> 00:09:27,500
And why is that?
185
00:09:27,500 --> 00:09:30,480
Well, notice that the graph of
'y' equals the absolute value
186
00:09:30,480 --> 00:09:33,640
of 'x' is the straight line
'y' equals 'x' if 'x' is
187
00:09:33,640 --> 00:09:37,040
non-negative and the straight
line 'y' equals 'minus x' if
188
00:09:37,040 --> 00:09:38,130
'x' is negative.
189
00:09:38,130 --> 00:09:41,960
In other words, 'f prime of x'
is 1 if 'x' is positive.
190
00:09:41,960 --> 00:09:44,940
It's minus 1 if 'x'
is negative.
191
00:09:44,940 --> 00:09:47,460
And hence, a jump discontinuity
in the
192
00:09:47,460 --> 00:09:49,840
derivative at 0.
193
00:09:49,840 --> 00:09:51,030
But you see, the
point is this.
194
00:09:51,030 --> 00:09:55,430
If you look at our graph,
do we have a low point
195
00:09:55,430 --> 00:09:57,020
at 'x' equals 0?
196
00:09:57,020 --> 00:09:59,030
In other words, is this the
lowest point in the
197
00:09:59,030 --> 00:10:01,180
neighborhood of 0?
198
00:10:01,180 --> 00:10:02,230
And the answer is yes.
199
00:10:02,230 --> 00:10:04,130
In fact, it's an 'absolute'
low point.
200
00:10:04,130 --> 00:10:07,030
Meaning that no matter where you
go, no point on our graph
201
00:10:07,030 --> 00:10:10,550
can be less or lower than this
particular point here.
202
00:10:10,550 --> 00:10:11,440
But that's irrelevant.
203
00:10:11,440 --> 00:10:12,350
The point is what?
204
00:10:12,350 --> 00:10:16,010
If we looked for a place where
'f prime of c' was 0 in this
205
00:10:16,010 --> 00:10:18,620
example, we wouldn't find one.
206
00:10:18,620 --> 00:10:20,520
That would not mean that there
wasn't a low point.
207
00:10:20,520 --> 00:10:21,880
What happened was what?
208
00:10:21,880 --> 00:10:24,220
The low point snuck in
at a place where the
209
00:10:24,220 --> 00:10:26,300
derivative did not exist.
210
00:10:26,300 --> 00:10:28,490
And again, you have
to be careful.
211
00:10:28,490 --> 00:10:29,750
All I said was what?
212
00:10:29,750 --> 00:10:34,210
Beware of points for which 'f
prime of c' does not exist.
213
00:10:34,210 --> 00:10:37,030
It does not mean that at each
point where 'f prime of c'
214
00:10:37,030 --> 00:10:39,030
does not exist that you're
going to have a
215
00:10:39,030 --> 00:10:40,270
high or a low point.
216
00:10:40,270 --> 00:10:44,090
For example, remember that the
derivative not existing
217
00:10:44,090 --> 00:10:46,440
loosely speaking means what?
218
00:10:46,440 --> 00:10:48,570
That there's a sharp corner
to the curve.
219
00:10:48,570 --> 00:10:50,510
What I'm thinking of is
something like this.
220
00:10:50,510 --> 00:10:52,520
Let's take a curve that's
always rising, say
221
00:10:52,520 --> 00:10:53,900
something like this.
222
00:10:53,900 --> 00:10:57,300
Now, let's suppose we put a
sharp corner in here, but in
223
00:10:57,300 --> 00:10:58,940
such a way that the curve still
224
00:10:58,940 --> 00:11:00,770
continues to always rise.
225
00:11:00,770 --> 00:11:03,180
Say like this.
226
00:11:03,180 --> 00:11:06,390
Now you see, at this particular
value of 'c', 'f
227
00:11:06,390 --> 00:11:11,860
prime of c' doesn't exist.
228
00:11:11,860 --> 00:11:13,010
I think you can see intuitively
229
00:11:13,010 --> 00:11:14,580
what's happening here.
230
00:11:14,580 --> 00:11:17,800
The slope approaches one value
as you approach 'c' from the
231
00:11:17,800 --> 00:11:22,040
left and another value if you
approach 'c' from the right.
232
00:11:22,040 --> 00:11:25,970
Now even know 'f prime of c'
doesn't exist at this point,
233
00:11:25,970 --> 00:11:27,550
notice that this point
is neither a
234
00:11:27,550 --> 00:11:29,170
high nor a low point.
235
00:11:29,170 --> 00:11:30,830
Meaning that all points what?
236
00:11:30,830 --> 00:11:34,710
To the left of this point are
below it and all points to the
237
00:11:34,710 --> 00:11:37,540
right of this point
are above it.
238
00:11:37,540 --> 00:11:38,460
So again, what?
239
00:11:38,460 --> 00:11:42,190
Beware when 'f prime of c'
equals 0, but don't jump to
240
00:11:42,190 --> 00:11:44,020
any false conclusions.
241
00:11:44,020 --> 00:11:48,450
And the third caution is an
extremely important one.
242
00:11:48,450 --> 00:11:51,450
In fact, for the first time
from a practical point of
243
00:11:51,450 --> 00:11:55,760
view, we are going to probably
see analytically what is the
244
00:11:55,760 --> 00:11:57,490
difference between
an open interval
245
00:11:57,490 --> 00:11:59,300
and a closed interval.
246
00:11:59,300 --> 00:12:03,190
Suppose I have the function 'f
of x' equals 'x squared', but
247
00:12:03,190 --> 00:12:06,810
the domain of 'f' is now the
open interval from 2 to 3.
248
00:12:06,810 --> 00:12:10,700
In other words, the inputs of
the 'f' machine are restricted
249
00:12:10,700 --> 00:12:13,850
to all those numbers which
are greater than 2,
250
00:12:13,850 --> 00:12:16,430
but less than 3.
251
00:12:16,430 --> 00:12:17,670
Now, let's take a look here.
252
00:12:17,670 --> 00:12:21,750
Let's first of all, see where
'f prime of x' is 0.
253
00:12:21,750 --> 00:12:26,610
First of all, 'f prime of x' is
'2x' and that equals 0 if
254
00:12:26,610 --> 00:12:29,480
and only if 'x' equals 0.
255
00:12:29,480 --> 00:12:31,880
Now here's where the domain
is very important.
256
00:12:31,880 --> 00:12:35,870
Is 'x' equal 0 in our domain?
257
00:12:35,870 --> 00:12:37,920
The answer is no.
258
00:12:37,920 --> 00:12:39,420
'x' equals 0 is not
in our domain.
259
00:12:39,420 --> 00:12:42,480
Our domain is restricted to be
the open interval from 2 to 3.
260
00:12:42,480 --> 00:12:45,660
Therefore, as far as our
function 'f' is concerned--
261
00:12:45,660 --> 00:12:48,790
and remember, way back in one
of our early, I don't even
262
00:12:48,790 --> 00:12:49,270
think it was a lecture.
263
00:12:49,270 --> 00:12:50,620
It was in our supplementary
notes.
264
00:12:50,620 --> 00:12:53,700
We pointed out that when you
define a function, you need
265
00:12:53,700 --> 00:12:57,510
not only the rule, but you
must specify the domain.
266
00:12:57,510 --> 00:13:00,110
Remember, two functions were
equal not only if they were
267
00:13:00,110 --> 00:13:04,250
the same rule, but they had to
be defined on the same domain.
268
00:13:04,250 --> 00:13:07,160
So f here is defined on
a domain from 2 to 3.
269
00:13:07,160 --> 00:13:10,380
And on that particular domain
of definition there is no
270
00:13:10,380 --> 00:13:12,750
place where the derivative
is 0.
271
00:13:12,750 --> 00:13:16,900
By the same token, since the
derivative is a polynomial and
272
00:13:16,900 --> 00:13:19,620
all polynomials are
differentiable, and all
273
00:13:19,620 --> 00:13:23,270
differentiable functions are
continuous, there certainly
274
00:13:23,270 --> 00:13:26,750
will be no places where 'f prime
of x' doesn't exist.
275
00:13:26,750 --> 00:13:34,140
In other words, 'f prime' exists
for all 'x' in the
276
00:13:34,140 --> 00:13:36,290
domain of 'f'.
277
00:13:36,290 --> 00:13:41,100
In other words, here is a
particular example in which a
278
00:13:41,100 --> 00:13:45,350
particular function on its
domain of definition does not
279
00:13:45,350 --> 00:13:48,500
have any high or low
points on it.
280
00:13:48,500 --> 00:13:52,100
And I'll illustrate that
graphically in a few moments.
281
00:13:52,100 --> 00:13:55,730
I simply wanted to put this on
the board first to put it into
282
00:13:55,730 --> 00:13:59,970
sharp contrast with what we're
going to say next.
283
00:13:59,970 --> 00:14:03,180
If we're not careful, in fact,
the next problem looks exactly
284
00:14:03,180 --> 00:14:05,340
the same as the problem
that we just solved.
285
00:14:05,340 --> 00:14:08,820
Namely, what we want to do now
is investigate the function 'f
286
00:14:08,820 --> 00:14:10,940
of x' equals 'x squared'.
287
00:14:10,940 --> 00:14:14,200
But we want the domain of
'f' now to be what?
288
00:14:14,200 --> 00:14:16,680
The closed interval
from 2 to 3.
289
00:14:16,680 --> 00:14:19,080
In other words, the only
difference between this
290
00:14:19,080 --> 00:14:22,300
problem and the problem that we
just solved is that now we
291
00:14:22,300 --> 00:14:26,190
want the endpoints included.
292
00:14:26,190 --> 00:14:28,720
Now, there's no sense
repeating the
293
00:14:28,720 --> 00:14:30,210
part that we did before.
294
00:14:30,210 --> 00:14:36,030
First of all, will 'f prime of
x' equal 0 in the domain of
295
00:14:36,030 --> 00:14:37,110
definition?
296
00:14:37,110 --> 00:14:38,990
As we saw before, no.
297
00:14:38,990 --> 00:14:41,650
298
00:14:41,650 --> 00:14:50,330
Is 'f prime' nonexistent any
place in the interval?
299
00:14:50,330 --> 00:14:51,400
No, it's differentiable.
300
00:14:51,400 --> 00:14:53,150
It's a smooth polynomial
curve.
301
00:14:53,150 --> 00:14:55,380
Answer is no.
302
00:14:55,380 --> 00:14:59,380
Now, here's where we come to two
very important points that
303
00:14:59,380 --> 00:15:02,410
hopefully, will clarify certain
conventions that were
304
00:15:02,410 --> 00:15:04,090
made in the textbook.
305
00:15:04,090 --> 00:15:07,420
In our section on continuity
there was a little result that
306
00:15:07,420 --> 00:15:09,370
may have seemed a little
bit obscure.
307
00:15:09,370 --> 00:15:10,300
It said what?
308
00:15:10,300 --> 00:15:14,190
That a continuous function
defined on a closed interval
309
00:15:14,190 --> 00:15:17,760
must take on its maximum and
minimum values someplace on
310
00:15:17,760 --> 00:15:19,360
that closed interval.
311
00:15:19,360 --> 00:15:22,630
You see, all we've proven
over here is what?
312
00:15:22,630 --> 00:15:33,640
That if 'f' has a max or min,
what we've proven is what?
313
00:15:33,640 --> 00:15:42,625
It does not occur in the open
interval from 'a' to 'b'.
314
00:15:42,625 --> 00:15:43,700
Well, look it.
315
00:15:43,700 --> 00:15:48,260
If the high and low points have
to occur some place on
316
00:15:48,260 --> 00:15:52,120
the closed interval and they
can't appear, as we've seen,
317
00:15:52,120 --> 00:15:56,960
in the open interval, where
must they occur?
318
00:15:56,960 --> 00:15:59,310
Well, if they can't be inside
and they have to be on the
319
00:15:59,310 --> 00:16:02,850
interval, it must be that they
take place at the endpoints.
320
00:16:02,850 --> 00:16:05,950
321
00:16:05,950 --> 00:16:07,590
Let's go back without referring
322
00:16:07,590 --> 00:16:08,920
back to another board.
323
00:16:08,920 --> 00:16:11,400
Remember we wrote down when
we're talking about high and
324
00:16:11,400 --> 00:16:15,370
low points that we talked about
'x' being in a delta
325
00:16:15,370 --> 00:16:16,890
neighborhood of 'c'.
326
00:16:16,890 --> 00:16:18,040
It meant what?
327
00:16:18,040 --> 00:16:23,390
That you could surround 'c'
by some bandwidth 'delta'.
328
00:16:23,390 --> 00:16:28,330
329
00:16:28,330 --> 00:16:30,870
And notice that our definition
in the textbook of a
330
00:16:30,870 --> 00:16:33,750
neighborhood was always
an open interval.
331
00:16:33,750 --> 00:16:36,440
And the reason that the
definition is given to be open
332
00:16:36,440 --> 00:16:39,420
is that notice that what the
definition now says is what?
333
00:16:39,420 --> 00:16:44,090
You know what's going on
on either side of 'c'.
334
00:16:44,090 --> 00:16:46,830
We know what's going on
either side of 'c'.
335
00:16:46,830 --> 00:16:52,050
Notice that in the case where
you have a closed interval, by
336
00:16:52,050 --> 00:16:54,530
definition of a closed interval
notice that what?
337
00:16:54,530 --> 00:17:00,220
We know what's happening as we
come in on 'a' from the right.
338
00:17:00,220 --> 00:17:03,450
And we know what's happening
to 'b' as we come in on it
339
00:17:03,450 --> 00:17:04,420
from the left.
340
00:17:04,420 --> 00:17:07,740
But allegedly, meaning that
since the function is only
341
00:17:07,740 --> 00:17:10,470
defined on the closed interval
from 'a' to 'b', we don't know
342
00:17:10,470 --> 00:17:13,150
what's happening before 'a',
and we don't know what's
343
00:17:13,150 --> 00:17:15,520
happening before 'b'.
344
00:17:15,520 --> 00:17:19,510
In other words, this is why the
test for 'f prime of c'
345
00:17:19,510 --> 00:17:25,250
equaling 0 applies only to 'c'
being in the interior.
346
00:17:25,250 --> 00:17:28,040
In other words, in
an open interval.
347
00:17:28,040 --> 00:17:31,030
So in other words then, you see
if a function happens to
348
00:17:31,030 --> 00:17:36,080
be continuous on the closed
interval and it doesn't have
349
00:17:36,080 --> 00:17:39,990
any high-low points in the
interior of the interval, then
350
00:17:39,990 --> 00:17:42,670
it must have its high-low
points where?
351
00:17:42,670 --> 00:17:44,720
It must be at the endpoints.
352
00:17:44,720 --> 00:17:47,460
And to show you quite simply
what was going on in this
353
00:17:47,460 --> 00:17:51,650
particular problem, notice
that if we graph 'f of x'
354
00:17:51,650 --> 00:17:55,830
equals 'x squared' and look at
this say, first of all, on the
355
00:17:55,830 --> 00:17:59,490
closed interval from 2 to 3 what
we're saying is look it,
356
00:17:59,490 --> 00:18:03,570
any point in which we look at a
neighborhood that isn't 2 or
357
00:18:03,570 --> 00:18:07,540
3, the curve is rising on
one side of the point.
358
00:18:07,540 --> 00:18:09,630
In other words, the curve is
rising on both sides of the
359
00:18:09,630 --> 00:18:13,030
point, so that if you are to
the left of the point, the
360
00:18:13,030 --> 00:18:15,700
height will be less than the
point you're interested in.
361
00:18:15,700 --> 00:18:18,410
If you're to the right of it,
the height will exceed the
362
00:18:18,410 --> 00:18:19,810
point that you're
interested in.
363
00:18:19,810 --> 00:18:23,260
But notice that at the end
points themselves, you do have
364
00:18:23,260 --> 00:18:24,420
what in this case?
365
00:18:24,420 --> 00:18:27,320
Not a relative high or
low, but actually an
366
00:18:27,320 --> 00:18:29,220
absolute high or low.
367
00:18:29,220 --> 00:18:31,570
In other words, the
point '2 comma 4'.
368
00:18:31,570 --> 00:18:34,890
See, what is the
endpoint here?
369
00:18:34,890 --> 00:18:37,800
One endpoint is 4, the other
endpoint going up on the
370
00:18:37,800 --> 00:18:39,440
y-direction here is 9.
371
00:18:39,440 --> 00:18:40,890
What you're saying is what?
372
00:18:40,890 --> 00:18:43,130
4 is less than 9.
373
00:18:43,130 --> 00:18:46,420
Every value of 'x squared'
between 2 and 3 falls
374
00:18:46,420 --> 00:18:48,190
between 4 and 9.
375
00:18:48,190 --> 00:18:49,230
And what you're saying
is what?
376
00:18:49,230 --> 00:18:51,800
That the lowest point occurs
when 'x' is 2, the highest
377
00:18:51,800 --> 00:18:54,090
point occurs when 'y' is 3.
378
00:18:54,090 --> 00:18:57,410
The interesting point to note is
that if you now look at the
379
00:18:57,410 --> 00:19:01,530
open interval and exclude the
endpoints, you cannot get a
380
00:19:01,530 --> 00:19:03,600
lowest point or a
highest point.
381
00:19:03,600 --> 00:19:06,960
Namely, notice that if you allow
yourself to get as close
382
00:19:06,960 --> 00:19:10,150
to 2 as you want without ever
getting there, it means you
383
00:19:10,150 --> 00:19:11,290
could have done what?
384
00:19:11,290 --> 00:19:12,880
Moved closer to 2.
385
00:19:12,880 --> 00:19:17,210
In other words, if 'x' is
greater than 2, you can pick
386
00:19:17,210 --> 00:19:19,190
another value that's what?
387
00:19:19,190 --> 00:19:20,270
Between 'x' and 2.
388
00:19:20,270 --> 00:19:21,700
There's a space there.
389
00:19:21,700 --> 00:19:23,240
In other words, what you saying
is that no matter how
390
00:19:23,240 --> 00:19:26,930
low you get here, as long as
you're not exactly at 2, you
391
00:19:26,930 --> 00:19:28,820
can find the point
that's lower.
392
00:19:28,820 --> 00:19:31,950
And in the same way as you move
out this way, as you move
393
00:19:31,950 --> 00:19:34,610
closer and closer to this point,
if you exclude this
394
00:19:34,610 --> 00:19:37,300
point itself, no matter where
you stop, you could have
395
00:19:37,300 --> 00:19:40,220
always found the point that
was a little bit higher.
396
00:19:40,220 --> 00:19:42,430
And again, you have
to be where?
397
00:19:42,430 --> 00:19:46,340
When I say check the endpoints
of a closed interval, it does
398
00:19:46,340 --> 00:19:48,070
not mean that the endpoints
will give
399
00:19:48,070 --> 00:19:49,710
you high or low points.
400
00:19:49,710 --> 00:19:52,670
For example, look at the
following curve.
401
00:19:52,670 --> 00:19:54,220
It looks something like this.
402
00:19:54,220 --> 00:19:55,250
It's continuous.
403
00:19:55,250 --> 00:19:58,010
It's defined on the closed
interval from 'a' to 'b'.
404
00:19:58,010 --> 00:20:02,710
Notice that at a we do not
get an absolute maximum.
405
00:20:02,710 --> 00:20:03,780
In fact, what?
406
00:20:03,780 --> 00:20:07,180
All of these points on the curve
are higher than what's
407
00:20:07,180 --> 00:20:08,480
happening here.
408
00:20:08,480 --> 00:20:09,790
Same thing happens, what?
409
00:20:09,790 --> 00:20:12,670
All of these points are lower
than what the height is
410
00:20:12,670 --> 00:20:14,500
corresponding to
'x' equals 'a'.
411
00:20:14,500 --> 00:20:17,250
And in the similar way, this
is what happens at 'b'.
412
00:20:17,250 --> 00:20:20,520
You see, with the endpoints,
obviously since you can't see
413
00:20:20,520 --> 00:20:23,710
what's happening before, this
will be either the highest
414
00:20:23,710 --> 00:20:27,180
point or the lowest point near
here depending on how the
415
00:20:27,180 --> 00:20:28,200
curve is sloped.
416
00:20:28,200 --> 00:20:29,550
But all we're saying is what?
417
00:20:29,550 --> 00:20:32,630
In terms of absolute high
values and absolute low
418
00:20:32,630 --> 00:20:35,500
values, meaning the highest
possible points and the lowest
419
00:20:35,500 --> 00:20:39,620
possible points, we must always
check the endpoints.
420
00:20:39,620 --> 00:20:42,970
But we can't be positive
that the endpoints
421
00:20:42,970 --> 00:20:45,450
are going to be chosen.
422
00:20:45,450 --> 00:20:47,070
In fact, let's summarize.
423
00:20:47,070 --> 00:20:48,750
And I'm going to summarize
again at
424
00:20:48,750 --> 00:20:49,540
the end of the lecture.
425
00:20:49,540 --> 00:20:50,830
But the idea is this.
426
00:20:50,830 --> 00:20:54,680
If 'f of x' is continuous on the
closed interval from 'a'
427
00:20:54,680 --> 00:20:56,800
to 'b'-- and here's
the key word.
428
00:20:56,800 --> 00:20:59,310
To find candidates--
429
00:20:59,310 --> 00:21:01,280
you know the old cliche
about many are
430
00:21:01,280 --> 00:21:02,930
called, but few are chosen.
431
00:21:02,930 --> 00:21:07,860
In this case, few are called
and even fewer are chosen.
432
00:21:07,860 --> 00:21:09,630
Namely, what we're saying
is look it.
433
00:21:09,630 --> 00:21:12,580
When we're looking for
high-low points, the
434
00:21:12,580 --> 00:21:13,700
candidates are what?
435
00:21:13,700 --> 00:21:17,410
Those points for which
the derivative is 0.
436
00:21:17,410 --> 00:21:20,550
We can check those out because
those are possibilities.
437
00:21:20,550 --> 00:21:23,030
Those points for which the
derivative fails to exist.
438
00:21:23,030 --> 00:21:26,220
We can check those out because
they're possibilities.
439
00:21:26,220 --> 00:21:27,470
And the endpoints.
440
00:21:27,470 --> 00:21:30,370
441
00:21:30,370 --> 00:21:32,030
Namely, if the function
is continuous,
442
00:21:32,030 --> 00:21:32,810
we check the endpoints.
443
00:21:32,810 --> 00:21:35,970
By the way, if the function is
not continuous, then there is
444
00:21:35,970 --> 00:21:37,780
no need to check-- well,
let's put it this way.
445
00:21:37,780 --> 00:21:40,190
If you're on an open interval
there's no need to check the
446
00:21:40,190 --> 00:21:42,440
endpoints because there
aren't any.
447
00:21:42,440 --> 00:21:44,260
In other words, notice that
I'm talking about what?
448
00:21:44,260 --> 00:21:48,190
That the function 'f' is not
only continuous, but on a
449
00:21:48,190 --> 00:21:49,620
closed interval.
450
00:21:49,620 --> 00:21:50,530
That's all there is to this.
451
00:21:50,530 --> 00:21:53,230
In other words, these are all
the possible candidates.
452
00:21:53,230 --> 00:21:54,920
Now you see the bigger
question is,
453
00:21:54,920 --> 00:21:56,500
how do you use this?
454
00:21:56,500 --> 00:21:59,140
And I thought that I would make
up a makeshift exercise,
455
00:21:59,140 --> 00:22:01,840
one that's rather easy to
do at the blackboard.
456
00:22:01,840 --> 00:22:06,370
For deeper exercises, for more
quantitative results, we have
457
00:22:06,370 --> 00:22:10,130
several exercises in the
learning exercises.
458
00:22:10,130 --> 00:22:11,610
Several exercises worked out
459
00:22:11,610 --> 00:22:13,320
illustratively in the textbook.
460
00:22:13,320 --> 00:22:17,250
But let's pick a particularly
straightforward example.
461
00:22:17,250 --> 00:22:24,430
Let's suppose that what I want
to do is construct a cylinder.
462
00:22:24,430 --> 00:22:26,510
This is a cross sectional
view of a cylinder.
463
00:22:26,510 --> 00:22:30,230
'x' represents the radius of
the base and 'y' represents
464
00:22:30,230 --> 00:22:31,290
the height.
465
00:22:31,290 --> 00:22:33,860
This is a right circular
cylinder.
466
00:22:33,860 --> 00:22:37,240
I'm given a constraint, namely
I'm told that for some reason
467
00:22:37,240 --> 00:22:40,350
or other, and I don't know why
anybody would ever want to
468
00:22:40,350 --> 00:22:43,140
impose this condition other
than the fact that we want
469
00:22:43,140 --> 00:22:45,160
some condition imposed here
to see what's happening.
470
00:22:45,160 --> 00:22:50,270
I'm told that I want the sum of
the radius of the base and
471
00:22:50,270 --> 00:22:53,040
the altitude to be exactly 30.
472
00:22:53,040 --> 00:22:55,400
In other words, if the radius
of the base is going to be 6
473
00:22:55,400 --> 00:22:58,440
inches, I want the altitude
to be 24 inches.
474
00:22:58,440 --> 00:22:59,360
That's a constraint.
475
00:22:59,360 --> 00:23:02,090
And by the way, I'll come
back to this later also.
476
00:23:02,090 --> 00:23:05,330
Notice how you're almost
begging a related rates
477
00:23:05,330 --> 00:23:07,050
relationship here.
478
00:23:07,050 --> 00:23:09,440
Or an implicit relationship
that 'x' and 'y' are not
479
00:23:09,440 --> 00:23:10,280
independent.
480
00:23:10,280 --> 00:23:12,275
But I've now put a constraint
on here.
481
00:23:12,275 --> 00:23:17,990
At any rate the question is,
how shall I use up my 30
482
00:23:17,990 --> 00:23:22,510
inches, say, if I want to make
the volume of the resulting
483
00:23:22,510 --> 00:23:25,080
cylinder as large as possible?
484
00:23:25,080 --> 00:23:26,450
And notice how we
work this thing.
485
00:23:26,450 --> 00:23:30,640
We say, well, the volume is
equal to 'pi 'x squared' y'.
486
00:23:30,640 --> 00:23:34,410
In this particular case, it's
easy to see explicitly that
487
00:23:34,410 --> 00:23:37,420
'y' is equal to '30 - x'.
488
00:23:37,420 --> 00:23:40,710
It's also easy to see physically
that 'x' must be
489
00:23:40,710 --> 00:23:41,620
more than 0.
490
00:23:41,620 --> 00:23:44,120
You can't have a negative
radius of the base.
491
00:23:44,120 --> 00:23:48,180
It must be less than 30 because
if you used up more
492
00:23:48,180 --> 00:23:51,080
than 30 inches in the radius
of your base, how can the
493
00:23:51,080 --> 00:23:54,970
radius of the base plus the
altitude add up to exactly 30?
494
00:23:54,970 --> 00:23:57,180
Because physically, the
constraint is that the
495
00:23:57,180 --> 00:23:59,100
altitude can't be negative.
496
00:23:59,100 --> 00:24:01,270
It's certainly a
positive value.
497
00:24:01,270 --> 00:24:03,340
So our analytic relationship
is what?
498
00:24:03,340 --> 00:24:06,060
That 'v' equals 'pi 'x squared'
y', which can be
499
00:24:06,060 --> 00:24:08,770
written as 'pi 'x squared'
times '30 - x''.
500
00:24:08,770 --> 00:24:12,290
Which in turn, can be written
as the polynomial '30 pi 'x
501
00:24:12,290 --> 00:24:15,750
squared'' minus 'pi 'x cubed'',
where 'x' is the open
502
00:24:15,750 --> 00:24:19,950
interval or defined on the open
interval from 0 to 30.
503
00:24:19,950 --> 00:24:21,730
Now, how do I proceed?
504
00:24:21,730 --> 00:24:24,480
What was my test
for membership?
505
00:24:24,480 --> 00:24:26,630
I have three ways of
checking out where
506
00:24:26,630 --> 00:24:28,380
high-low points will occur.
507
00:24:28,380 --> 00:24:31,190
Or max-min points in this case,
since notice that this
508
00:24:31,190 --> 00:24:34,750
function does not require a
graph to understand it.
509
00:24:34,750 --> 00:24:38,240
I first check out to see where
the derivative is 0.
510
00:24:38,240 --> 00:24:41,090
Well, the derivative
is simply '60 pi x'
511
00:24:41,090 --> 00:24:43,210
minus '3 pi 'x squared''.
512
00:24:43,210 --> 00:24:48,250
If I set this thing equal to 0,
I find upon factoring that
513
00:24:48,250 --> 00:24:52,090
either 'x' must be 0
or 'x' must be 20.
514
00:24:52,090 --> 00:24:58,520
I can immediately exclude 'x'
equals 0 because notice that
515
00:24:58,520 --> 00:25:02,800
my function 'v' had its domain
of definition on the open
516
00:25:02,800 --> 00:25:04,430
interval from 0 to 30.
517
00:25:04,430 --> 00:25:07,690
'x' equals 0 is not in the open
interval from 0 to 30.
518
00:25:07,690 --> 00:25:10,190
Consequently, we can rule
this thing out.
519
00:25:10,190 --> 00:25:13,390
And what we discover is
that 'x' equals 20.
520
00:25:13,390 --> 00:25:15,980
So 'x' equals 20 is
the only possible
521
00:25:15,980 --> 00:25:17,410
candidate that we have.
522
00:25:17,410 --> 00:25:20,650
And by the way, since the sum of
'x' and 'y' must be 20, if
523
00:25:20,650 --> 00:25:23,240
'x' equals 20, 'y' must be 10.
524
00:25:23,240 --> 00:25:25,715
So the only candidate that we
have by setting the derivative
525
00:25:25,715 --> 00:25:29,330
equal to 0 is that 'x' should
be 20 and 'y' should be 10.
526
00:25:29,330 --> 00:25:32,540
We do not get any candidates
in the sense of where the
527
00:25:32,540 --> 00:25:33,890
derivative doesn't exist.
528
00:25:33,890 --> 00:25:37,250
Because if you look at '60 pi
x' minus '3 pi 'x squared'',
529
00:25:37,250 --> 00:25:40,170
this certainly exists for
all values of 'x'.
530
00:25:40,170 --> 00:25:43,410
And finally, we have no
endpoints to check.
531
00:25:43,410 --> 00:25:46,800
Because again, our
function 'v' is
532
00:25:46,800 --> 00:25:50,310
defined on an open interval.
533
00:25:50,310 --> 00:25:54,170
By the way, if I tried to draw
this somewhat to scale, a very
534
00:25:54,170 --> 00:25:58,690
interesting result turns up,
which may show the power of
535
00:25:58,690 --> 00:26:03,280
analytical methods versus more
intuitive types of methods.
536
00:26:03,280 --> 00:26:05,220
You see, what we showed
here was what?
537
00:26:05,220 --> 00:26:07,190
Without going into the details,
that to get the
538
00:26:07,190 --> 00:26:11,760
largest volume cylinder,
'x' should be 20.
539
00:26:11,760 --> 00:26:14,210
In other words, the radius of
your base should be 20 and the
540
00:26:14,210 --> 00:26:15,750
altitude should be 10.
541
00:26:15,750 --> 00:26:19,240
Let's take a look at that
drawn roughly to scale.
542
00:26:19,240 --> 00:26:22,530
The radius of the base here is
20 and the altitude is 10.
543
00:26:22,530 --> 00:26:23,800
By the way, that means
that the cross
544
00:26:23,800 --> 00:26:25,160
section will be what?
545
00:26:25,160 --> 00:26:30,430
A rectangle whose height is
10 and whose base is 40.
546
00:26:30,430 --> 00:26:33,670
Now intuitively, I think it's
easy to see that for a
547
00:26:33,670 --> 00:26:39,100
rectangle, for a given perimeter
the largest possible
548
00:26:39,100 --> 00:26:42,880
area rectangle is the one
which is a square.
549
00:26:42,880 --> 00:26:45,690
Well, without even trying to
prove that, let's go to this
550
00:26:45,690 --> 00:26:46,150
thing here.
551
00:26:46,150 --> 00:26:49,370
Instead of using up the 20 and
the 10, let's draw a second
552
00:26:49,370 --> 00:26:52,940
rectangle, or a second cross
section of a cylinder where
553
00:26:52,940 --> 00:26:56,350
the radius is 15 and
the height is 15.
554
00:26:56,350 --> 00:26:58,860
Notice that this still satisfies
the fact that the
555
00:26:58,860 --> 00:27:02,520
sum of the radius and
the altitude are 30.
556
00:27:02,520 --> 00:27:03,890
Now, look at this.
557
00:27:03,890 --> 00:27:08,220
If we compute the area of this
particular rectangle, 40 times
558
00:27:08,220 --> 00:27:10,350
10, it's 400.
559
00:27:10,350 --> 00:27:12,490
On the other hand, the
volume is what?
560
00:27:12,490 --> 00:27:14,380
Pi times 20 squared.
561
00:27:14,380 --> 00:27:15,830
The radius of the
base squared.
562
00:27:15,830 --> 00:27:17,320
Times the height, which is 10.
563
00:27:17,320 --> 00:27:20,890
And that yields the result
of 4,000 pi.
564
00:27:20,890 --> 00:27:24,540
On the other hand, if we look
at our second rectangle, its
565
00:27:24,540 --> 00:27:27,890
area is 450.
566
00:27:27,890 --> 00:27:29,240
But its volume is what?
567
00:27:29,240 --> 00:27:37,270
It's pi times 15 squared times
15, which is 3,375 pi.
568
00:27:37,270 --> 00:27:40,760
The interesting result
here is what?
569
00:27:40,760 --> 00:27:45,050
The area of this rectangle
is greater than the
570
00:27:45,050 --> 00:27:48,280
area of this rectangle.
571
00:27:48,280 --> 00:27:52,430
In fact, the area of the second
rectangle is 450.
572
00:27:52,430 --> 00:27:54,800
The area of the first rectangle
is only 400.
573
00:27:54,800 --> 00:27:58,840
But notice that when you revolve
this thing to form the
574
00:27:58,840 --> 00:28:05,170
cylinder, the smaller cross
sectional area generates the
575
00:28:05,170 --> 00:28:06,440
larger volume.
576
00:28:06,440 --> 00:28:09,340
And the reason, of course, for
that is that the relationship
577
00:28:09,340 --> 00:28:11,840
in our variables
was not linear.
578
00:28:11,840 --> 00:28:14,860
In other words notice that
when 'x' is large, a
579
00:28:14,860 --> 00:28:18,340
relatively small change in
'x' produces a large
580
00:28:18,340 --> 00:28:20,760
change in 'x squared'.
581
00:28:20,760 --> 00:28:24,490
In other words, a relatively
small value in 'x' can offset
582
00:28:24,490 --> 00:28:27,740
a relatively large value
or increase in 'y'.
583
00:28:27,740 --> 00:28:29,840
And this is kind of interesting
because try to
584
00:28:29,840 --> 00:28:33,040
figure out intuitively how you
would figure out where these
585
00:28:33,040 --> 00:28:34,880
stop compensating
for one another?
586
00:28:34,880 --> 00:28:37,350
Where does it turn out that
finally you've taken so much
587
00:28:37,350 --> 00:28:41,100
away from 'x' that even though
you square it, it can't
588
00:28:41,100 --> 00:28:42,940
compensate the change in 'y'?
589
00:28:42,940 --> 00:28:46,050
How would you intuitively pick
off where the high-low points
590
00:28:46,050 --> 00:28:47,900
occur in a problem like this?
591
00:28:47,900 --> 00:28:51,080
And all I want you to see is
again, the beautiful gentle
592
00:28:51,080 --> 00:28:53,840
balance between intuitive
calculus
593
00:28:53,840 --> 00:28:55,570
and rigorous calculus.
594
00:28:55,570 --> 00:28:57,460
That we don't throw away
our intuition.
595
00:28:57,460 --> 00:29:01,160
But notice how, in many cases,
where our intuition fails us,
596
00:29:01,160 --> 00:29:04,190
the analytic recipes
come to our rescue.
597
00:29:04,190 --> 00:29:07,550
But enough said about that, let
me now again, highlight
598
00:29:07,550 --> 00:29:10,840
the difference or the
relationship between functions
599
00:29:10,840 --> 00:29:11,850
and graphs.
600
00:29:11,850 --> 00:29:14,660
Namely, in all of this
discussion that we've done on
601
00:29:14,660 --> 00:29:17,510
this particular board, we're
talking about what?
602
00:29:17,510 --> 00:29:19,330
'v' being a function of 'x'.
603
00:29:19,330 --> 00:29:22,810
We do not have to visualize
this thing pictorially.
604
00:29:22,810 --> 00:29:28,140
But if we wish, what we can say
is let's graph 'v' as a
605
00:29:28,140 --> 00:29:30,560
function of 'x'.
606
00:29:30,560 --> 00:29:33,520
And you see, going back to the
material of last time.
607
00:29:33,520 --> 00:29:36,560
And notice, you see how
interrelated these things are.
608
00:29:36,560 --> 00:29:39,660
Notice how curve plotting
ties in very nicely with
609
00:29:39,660 --> 00:29:40,740
derivatives and the like.
610
00:29:40,740 --> 00:29:42,120
All we're saying is what?
611
00:29:42,120 --> 00:29:45,920
Given this relationship, which
is how 'v' is related to 'x',
612
00:29:45,920 --> 00:29:48,530
we can form the first
derivative, we can form the
613
00:29:48,530 --> 00:29:51,350
second derivative, we can look
to see where the first
614
00:29:51,350 --> 00:29:53,880
derivative is 0, we can look
to see where the second
615
00:29:53,880 --> 00:29:54,960
derivative is 0.
616
00:29:54,960 --> 00:29:57,980
I leave these details to you
because after the homework
617
00:29:57,980 --> 00:30:01,370
assignment you did last time
these should be fairly trivial
618
00:30:01,370 --> 00:30:02,970
exercises to do.
619
00:30:02,970 --> 00:30:05,580
But the idea is if you now
utilize all of this
620
00:30:05,580 --> 00:30:09,300
information, you find that if
you plot 'v' verses 'x', you
621
00:30:09,300 --> 00:30:12,190
get a graph something
like this.
622
00:30:12,190 --> 00:30:16,330
By the way, again notice that we
were not talking about just
623
00:30:16,330 --> 00:30:18,410
'v' being a function of 'x'.
624
00:30:18,410 --> 00:30:22,010
The domain of 'v' was restricted
to be the open
625
00:30:22,010 --> 00:30:25,420
interval from 0 to 30, and this
is a very crucial thing
626
00:30:25,420 --> 00:30:26,580
to keep in mind.
627
00:30:26,580 --> 00:30:29,120
You can get into a whole bunch
of trouble if you start
628
00:30:29,120 --> 00:30:31,210
looking to see what
happens out here.
629
00:30:31,210 --> 00:30:33,595
For example, you say, hey, this
curve is always going to
630
00:30:33,595 --> 00:30:34,240
keep going up.
631
00:30:34,240 --> 00:30:37,210
Won't this be greater than this
maximum value over here
632
00:30:37,210 --> 00:30:38,040
eventually?
633
00:30:38,040 --> 00:30:39,600
The answer is yes, it will be.
634
00:30:39,600 --> 00:30:42,600
But what does it mean to say
that 'x' is negative?
635
00:30:42,600 --> 00:30:44,500
'x' was the radius
of our base.
636
00:30:44,500 --> 00:30:46,070
So in other words,
what we should
637
00:30:46,070 --> 00:30:48,360
really say here is what?
638
00:30:48,360 --> 00:30:50,930
That the function that we're
talking about is not this
639
00:30:50,930 --> 00:30:51,930
whole curve.
640
00:30:51,930 --> 00:30:54,470
Yikes, that wasn't a very
good job of drawing.
641
00:30:54,470 --> 00:30:55,560
But rather what?
642
00:30:55,560 --> 00:30:59,370
Just this portion of the curve
defined on the open interval
643
00:30:59,370 --> 00:31:01,420
from 0 to 30.
644
00:31:01,420 --> 00:31:05,120
By the way, let me point
out something else.
645
00:31:05,120 --> 00:31:07,690
You recall in an earlier
lecture we talked about
646
00:31:07,690 --> 00:31:08,960
related rates.
647
00:31:08,960 --> 00:31:11,130
We had an assignment with other
lecture where you've
648
00:31:11,130 --> 00:31:13,420
solved some problems using
related rates.
649
00:31:13,420 --> 00:31:16,690
Let me show you how related
rates play a very important
650
00:31:16,690 --> 00:31:21,300
computational role in dealing
with max-min problems.
651
00:31:21,300 --> 00:31:24,780
In this particular problem, we
had that 'v' equals 'pi 'x
652
00:31:24,780 --> 00:31:28,240
squared' y', where 'y' happened
to be a particular
653
00:31:28,240 --> 00:31:29,300
function of 'x'.
654
00:31:29,300 --> 00:31:33,270
In fact, implicitly it was given
by the fact that 'x + y'
655
00:31:33,270 --> 00:31:34,970
happened to equal 30.
656
00:31:34,970 --> 00:31:36,730
Now let's keep track
of something here.
657
00:31:36,730 --> 00:31:40,220
In this particular problem,
notice that it was very easy
658
00:31:40,220 --> 00:31:42,550
to change this implicit
relationship
659
00:31:42,550 --> 00:31:44,180
to an explicit one.
660
00:31:44,180 --> 00:31:48,520
It was also easy once you
expressed 'y' explicitly in
661
00:31:48,520 --> 00:31:51,120
terms of 'x' that when you
wanted to substitute into
662
00:31:51,120 --> 00:31:54,950
here, it was a very easy
computational job to carry out
663
00:31:54,950 --> 00:31:56,080
the operations.
664
00:31:56,080 --> 00:31:59,430
But suppose there happened to be
cube roots in here, or all
665
00:31:59,430 --> 00:32:02,380
sorts of nasty things, whatever
they might be.
666
00:32:02,380 --> 00:32:05,910
And suppose instead of 'x + y'
equals 30, you had our old
667
00:32:05,910 --> 00:32:08,870
friend something like 'x to the
eighth' plus ''x to the
668
00:32:08,870 --> 00:32:13,560
sixth' 'y squared'' plus 'y
to the sixth' equals 3.
669
00:32:13,560 --> 00:32:15,530
How would you solve for
'y' explicitly in
670
00:32:15,530 --> 00:32:16,870
terms of 'x' there?
671
00:32:16,870 --> 00:32:18,970
And the point that I'd like
you to see is that we can
672
00:32:18,970 --> 00:32:23,570
solve this problem very nicely
without having to resort to
673
00:32:23,570 --> 00:32:26,460
explicitly replacing 'y'
as a function of 'x'.
674
00:32:26,460 --> 00:32:30,130
Namely, implicitly assuming that
'y' is a differentiable
675
00:32:30,130 --> 00:32:32,270
function of 'x', we
can differentiate
676
00:32:32,270 --> 00:32:33,760
this thing as a product.
677
00:32:33,760 --> 00:32:35,320
'dv dx' will be what?
678
00:32:35,320 --> 00:32:38,280
The derivative of the first
factor, which is '2 pi x',
679
00:32:38,280 --> 00:32:39,690
times the second.
680
00:32:39,690 --> 00:32:42,970
Plus the first factor, which is
'pi 'x squared', times the
681
00:32:42,970 --> 00:32:45,730
derivative of 'y' with
respect to 'x'.
682
00:32:45,730 --> 00:32:47,440
So that's 'dv dx'.
683
00:32:47,440 --> 00:32:50,960
On the other hand, from this
relationship here, we can
684
00:32:50,960 --> 00:32:54,380
conclude by differentiating
implicitly that '1
685
00:32:54,380 --> 00:32:57,810
+ 'dy dx'' is 0.
686
00:32:57,810 --> 00:33:03,550
And therefore, 'dy
dx' is minus 1.
687
00:33:03,550 --> 00:33:08,130
And putting that value for 'dy
dx' in here, we wind up with
688
00:33:08,130 --> 00:33:09,920
this explicit relationship.
689
00:33:09,920 --> 00:33:14,340
And we can now see that 'dv dx'
is 0 if and only if just
690
00:33:14,340 --> 00:33:15,710
by solving this thing,
setting it equal to
691
00:33:15,710 --> 00:33:17,870
0, 'x' equals '2y'.
692
00:33:17,870 --> 00:33:20,400
By the way, that's exactly what
happened in our problem.
693
00:33:20,400 --> 00:33:23,340
You'll notice that 'x' turned
out to be 20 and 'y' turned
694
00:33:23,340 --> 00:33:25,050
out to be 10.
695
00:33:25,050 --> 00:33:27,370
You see again, 'x' and
'y' are related.
696
00:33:27,370 --> 00:33:31,060
You can say, gee, couldn't
'x' be 60 and 'y' be 30?
697
00:33:31,060 --> 00:33:32,370
Answer, no.
698
00:33:32,370 --> 00:33:36,440
Because the constraint is
that 'x + y' must be 30.
699
00:33:36,440 --> 00:33:37,250
Well, look it.
700
00:33:37,250 --> 00:33:40,820
I don't want us to get too
wrapped up on the idea of
701
00:33:40,820 --> 00:33:42,330
computational differences now.
702
00:33:42,330 --> 00:33:47,320
What I do want to do is review
our basic result.
703
00:33:47,320 --> 00:33:48,660
And in fact, let me
come over here.
704
00:33:48,660 --> 00:33:50,180
I hope this doesn't spoil you.
705
00:33:50,180 --> 00:33:51,750
I'd like to come back
to the board.
706
00:33:51,750 --> 00:33:54,330
And actually, since I've got
this all written down, let's
707
00:33:54,330 --> 00:33:57,180
close on this particular
result again.
708
00:33:57,180 --> 00:34:01,170
To find the high-low points of
a function 'f of x', we first
709
00:34:01,170 --> 00:34:04,700
of all, check out when 'f
prime of c' is 0 for
710
00:34:04,700 --> 00:34:05,350
candidates.
711
00:34:05,350 --> 00:34:08,790
We check out where 'f prime
of c' fails to exist.
712
00:34:08,790 --> 00:34:12,460
And if it's a closed interval,
we check the endpoints.
713
00:34:12,460 --> 00:34:15,480
This is the mechanism behind
what we're doing from that
714
00:34:15,480 --> 00:34:19,370
point on, as the cliche goes,
it's all engineering's baby.
715
00:34:19,370 --> 00:34:21,900
It's all computational
know how.
716
00:34:21,900 --> 00:34:25,460
At any rate, I think this is
enough in terms of emphasizing
717
00:34:25,460 --> 00:34:27,060
the points that we
wanted to make.
718
00:34:27,060 --> 00:34:29,120
And so until next
time, goodbye.
719
00:34:29,120 --> 00:34:32,199
720
00:34:32,199 --> 00:34:34,739
ANNOUNCER: Funding for the
publication of this video was
721
00:34:34,739 --> 00:34:39,449
provided by the Gabriella and
Paul Rosenbaum Foundation.
722
00:34:39,449 --> 00:34:43,630
Help OCW continue to provide
free and open access to MIT
723
00:34:43,630 --> 00:34:47,820
courses by making a donation
at ocw.mit.edu/donate.
724
00:34:47,820 --> 00:34:52,564