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So, today we are going to
continue looking at critical
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points,
and we'll learn how to actually
00:00:31.000 --> 00:00:33.000
decide whether a typical point
is a minimum,
00:00:33.000 --> 00:00:37.000
maximum, or a saddle point.
So, that's the main topic for
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today.
So, remember yesterday,
00:00:41.000 --> 00:00:50.000
we looked at critical points of
functions of several variables.
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And, so a critical point
functions, we have two values,
00:00:58.000 --> 00:01:05.000
x and y.
That's a point where the
00:01:05.000 --> 00:01:11.000
partial derivatives are both
zero.
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And, we've seen that there's
various kinds of critical
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points.
There's local minima.
00:01:20.000 --> 00:01:24.000
So, maybe I should show the
function on this contour
00:01:24.000 --> 00:01:28.000
plot,there is local maxima,
which are like that.
00:01:28.000 --> 00:01:35.000
And, there's saddle points
which are neither minima nor
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maxima.
And, of course,
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if you have a real function,
then it would be more
00:01:41.000 --> 00:01:45.000
complicated.
It will have several critical
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points.
So, this example here,
00:01:48.000 --> 00:01:54.000
well, you see on the plot that
there is two maxima.
00:01:54.000 --> 00:01:58.000
And, there is in the middle,
between them,
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a saddle point.
And, actually,
00:02:00.000 --> 00:02:02.000
you can see them on the contour
plot.
00:02:02.000 --> 00:02:07.000
On the contour plot,
you see the maxima because the
00:02:07.000 --> 00:02:12.000
level curves become circles that
now down and shrink to the
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maximum.
And, you can see the saddle
00:02:15.000 --> 00:02:18.000
point because here you have this
level curve that makes a figure
00:02:18.000 --> 00:02:20.000
eight.
It crosses itself.
00:02:20.000 --> 00:02:25.000
And, if you move up or down
here, so along the y direction,
00:02:25.000 --> 00:02:28.000
the values of the function will
decrease.
00:02:28.000 --> 00:02:32.000
Along the x direction,
the values will increase.
00:02:32.000 --> 00:02:37.000
So, you can see usually quite
easily where are the critical
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points just by looking either at
the graph or at the contour
00:02:42.000 --> 00:02:44.000
plots.
So, the only thing with the
00:02:44.000 --> 00:02:47.000
contour plots is you need to
read the values to tell a
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minimum from a maximum because
the contour plots look the same.
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Just, of course,
in one case,
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the values increase,
and in another one they
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decrease.
So, the question -- -- is,
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how do we decide -- -- between
the various possibilities?
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So, local minimum,
local maximum,
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or saddle point.
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So, and, in fact,
why do we care?
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Well, the other question is how
do we find the global
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minimum/maximum of a function?
So, here what I should point
00:04:05.000 --> 00:04:07.000
out, well,
first of all,
00:04:07.000 --> 00:04:09.000
to decide where the function is
the largest,
00:04:09.000 --> 00:04:12.000
in general you'll have actually
to compare the values.
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For example,
here, if you want to know,
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what is the maximum of this
function?
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Well, we have two obvious
candidates.
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We have this local maximum and
that local maximum.
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And, the question is,
which one is the higher of the
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two?
Well, in this case,
00:04:26.000 --> 00:04:30.000
actually, there is actually a
tie for maximum.
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But, in general,
you would have to compute the
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function at both points,
and compare the values if you
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know that it's three at one of
them and four at the other.
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Well, four wins.
The other thing that you see
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here is if you are looking for
the minimum of this function,
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well, the minimum is not going
to be at any of the critical
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points.
So, where's the minimum?
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Well, it looks like the minimum
is actually out there on the
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boundary or at infinity.
So, that's another feature.
00:04:56.000 --> 00:04:59.000
The global minimum or maximum
doesn't have to be at a critical
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point.
It could also be,
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somehow, on the side in some
limiting situation where one
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variable stops being in the
allowed rang of values or goes
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to infinity.
So, we have to actually check
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the boundary and the infinity
behavior of our function to know
00:05:19.000 --> 00:05:23.000
where, actually,
the minimum and maximum will
00:05:23.000 --> 00:05:27.000
be.
So, in general,
00:05:27.000 --> 00:05:37.000
I should point out,
these should occur either at
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the critical point or on the
boundary or at infinity.
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So, by that,
I mean on the boundary of a
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domain of definition that we are
considering.
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And so, we have to try both.
OK, but so we'll get back to
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that.
For now, let's try to focus on
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the question of,
you know, what's the type of
00:06:09.000 --> 00:06:16.000
the critical point?
So, we'll use something that's
00:06:16.000 --> 00:06:21.000
known as the second derivative
test.
00:06:21.000 --> 00:06:25.000
And, in principle,
well, the idea is kind of
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similar to what you do with the
function of one variable,
00:06:29.000 --> 00:06:32.000
namely, the function of one
variable.
00:06:32.000 --> 00:06:34.000
If the derivative is zero,
then you know that you should
00:06:34.000 --> 00:06:38.000
look at the second derivative.
And, that will tell you whether
00:06:38.000 --> 00:06:41.000
it's curving up or down whether
you have a local max and the
00:06:41.000 --> 00:06:44.000
local min.
And, the main problem here is,
00:06:44.000 --> 00:06:46.000
of course, we have more
possible situations,
00:06:46.000 --> 00:06:48.000
and we have several
derivatives.
00:06:48.000 --> 00:06:52.000
So, we have to think a bit
harder about how we'll decide.
00:06:52.000 --> 00:06:56.000
But, it will again involve the
second derivative.
00:06:56.000 --> 00:07:01.000
OK, so let's start with just an
easy example that will be useful
00:07:01.000 --> 00:07:06.000
to us because actually it will
provide the basis for the
00:07:06.000 --> 00:07:10.000
general method.
OK, so we are first going to
00:07:10.000 --> 00:07:15.000
consider a case where we have a
function that's actually just
00:07:15.000 --> 00:07:20.000
quadratic.
So, let's say I have a
00:07:20.000 --> 00:07:28.000
function, W of (x,y) that's of
the form ax^2 bxy cy^2.
00:07:28.000 --> 00:07:32.000
OK, so this guy has a critical
point at the origin because if
00:07:32.000 --> 00:07:36.000
you take the derivative with
respect to x,
00:07:36.000 --> 00:07:38.000
well, and if you plug x equals
y equals zero,
00:07:38.000 --> 00:07:42.000
you'll get zero,
and same with respect to y.
00:07:42.000 --> 00:07:44.000
You can also see,
if you try to do a linear
00:07:44.000 --> 00:07:47.000
approximation of this,
well, all these guys are much
00:07:47.000 --> 00:07:50.000
smaller than x and y when x and
y are small.
00:07:50.000 --> 00:07:55.000
So, the linear approximation,
the tangent plane to the graph
00:07:55.000 --> 00:07:59.000
is really just w=0.
OK, so, how do we do it?
00:07:59.000 --> 00:08:03.000
Well, yesterday we actually did
an example.
00:08:03.000 --> 00:08:09.000
It was a bit more complicated
than that, but let me do it,
00:08:09.000 --> 00:08:13.000
so remember,
we were looking at something
00:08:13.000 --> 00:08:19.000
that started with x^2 2xy 3y^2.
And, there were other terms.
00:08:19.000 --> 00:08:23.000
But, let's forget them now.
And, what we did is we said,
00:08:23.000 --> 00:08:28.000
well, we can rewrite this as (x
y)^2 2y^2.
00:08:28.000 --> 00:08:31.000
And now, this is a sum of two
squares.
00:08:31.000 --> 00:08:35.000
So, each of these guys has to
be nonnegative.
00:08:35.000 --> 00:08:40.000
And so, the origin will be a
minimum.
00:08:40.000 --> 00:08:44.000
Well, it turns out we can do
something similar in general no
00:08:44.000 --> 00:08:47.000
matter what the values of a,
b, and c are.
00:08:47.000 --> 00:08:50.000
We'll just try to first
complete things to a square.
00:08:50.000 --> 00:08:55.000
OK, so let's do that.
So, in general,
00:08:55.000 --> 00:09:01.000
well, let me be slightly less
general, and let me assume that
00:09:01.000 --> 00:09:08.000
a is not zero because otherwise
I can't do what I'm going to do.
00:09:08.000 --> 00:09:20.000
So, I'm going to write this as
a times x^2 plus b over axy.
00:09:20.000 --> 00:09:25.000
And then I have my cy^2.
And now this looks like the
00:09:25.000 --> 00:09:28.000
beginning of the square of
something, OK,
00:09:28.000 --> 00:09:31.000
just like what we did over
there.
00:09:31.000 --> 00:09:39.000
So, what is it the square of?
Well, you'd start with x plus I
00:09:39.000 --> 00:09:45.000
claim if I put b over 2a times y
and I square it,
00:09:45.000 --> 00:09:52.000
then see the cross term two
times x times b over 2a y will
00:09:52.000 --> 00:09:57.000
become b over axy.
Of course, now I also get some
00:09:57.000 --> 00:10:01.000
y squares out of this.
How many y squares do I get?
00:10:01.000 --> 00:10:05.000
Well, I get b^2 over 4a^2 times
a.
00:10:05.000 --> 00:10:11.000
So, I get b2 over 4a y^2.
So, and I want,
00:10:11.000 --> 00:10:17.000
in fact, c times y^2.
So, the number of y^2 that I
00:10:17.000 --> 00:10:22.000
should add is c minus b^2 over
4a.
00:10:22.000 --> 00:10:27.000
OK, let's see that again.
If I expand this thing,
00:10:27.000 --> 00:10:33.000
I will get ax^2 plus a times b
over 2a times 2xy.
00:10:33.000 --> 00:10:39.000
That's going to be my bxy.
But, I also get b^2 over 4a^2
00:10:39.000 --> 00:10:44.000
y^2 times a.
That's b^2 over 4ay^2.
00:10:44.000 --> 00:10:47.000
And, that cancels out with this
guy here.
00:10:47.000 --> 00:10:52.000
And then, I will be left with
cy^2.
00:10:52.000 --> 00:10:58.000
OK, do you see it kind of?
OK, if not, well,
00:10:58.000 --> 00:11:04.000
try expanding this square
again.
00:11:04.000 --> 00:11:06.000
OK, maybe I'll do it just to
convince you.
00:11:06.000 --> 00:11:11.000
But, so if I expand this,
I will get A times,
00:11:11.000 --> 00:11:16.000
let me put that in a different
color because you shouldn't
00:11:16.000 --> 00:11:19.000
write that down.
It's just to convince you again.
00:11:19.000 --> 00:11:25.000
So, if you don't see it yet,
let's expend this thing.
00:11:25.000 --> 00:11:35.000
We'll get a times x^2 plus a
times 2xb over 2ay.
00:11:35.000 --> 00:11:42.000
Well, the two A's cancel out.
We get bxy plus a times the
00:11:42.000 --> 00:11:53.000
square of that's going to be b^2
over 4a^2 y^2 plus cy^2 minus
00:11:53.000 --> 00:11:59.000
b^2 over 4ay^2.
Here, the a and the a
00:11:59.000 --> 00:12:06.000
simplifies, and now these two
terms simplify and give me just
00:12:06.000 --> 00:12:09.000
cy^2 in the end.
OK, and that's kind of
00:12:09.000 --> 00:12:12.000
unreadable after I've canceled
everything,
00:12:12.000 --> 00:12:19.000
but if you follow it,
you see that basically I've
00:12:19.000 --> 00:12:24.000
just rewritten my initial
function.
00:12:24.000 --> 00:12:29.000
OK, is that kind of OK?
I mean, otherwise there's just
00:12:29.000 --> 00:12:32.000
no substitute.
You'll have to do it yourself,
00:12:32.000 --> 00:12:38.000
I'm afraid.
OK, so, let me continue to play
00:12:38.000 --> 00:12:43.000
with this.
So, I'm just going to put this
00:12:43.000 --> 00:12:48.000
in a slightly different form
just to clear the denominators.
00:12:48.000 --> 00:12:56.000
OK, so, I will instead write
this as one over 4a times the
00:12:56.000 --> 00:13:03.000
big thing.
So, I'm going to just put 4a^2
00:13:03.000 --> 00:13:10.000
times x plus b over 2ay squared.
OK, so far I have the same
00:13:10.000 --> 00:13:13.000
thing as here.
I just introduced the 4a that
00:13:13.000 --> 00:13:19.000
cancels out, plus for the other
one, I'm just clearing the
00:13:19.000 --> 00:13:28.000
denominator.
I end up with (4ac-b^2)y^2.
00:13:28.000 --> 00:13:32.000
OK, so that's a lot of terms.
But, what does it look like?
00:13:32.000 --> 00:13:35.000
Well, it looks like,
so we have some constant
00:13:35.000 --> 00:13:38.000
factors, and here we have a
square, and here we have a
00:13:38.000 --> 00:13:39.000
square.
So, basically,
00:13:39.000 --> 00:13:44.000
we've written this as a sum of
two squares, well,
00:13:44.000 --> 00:13:47.000
a sum or a difference of two
squares.
00:13:47.000 --> 00:13:51.000
And, maybe that's what we need
to figure out to know what kind
00:13:51.000 --> 00:13:55.000
of point it is because,
see, if you take a sum of two
00:13:55.000 --> 00:13:57.000
squares,
that you will know that each
00:13:57.000 --> 00:14:01.000
square takes nonnegative values.
And you will have,
00:14:01.000 --> 00:14:04.000
the function will always take
nonnegative values.
00:14:04.000 --> 00:14:07.000
So, the origin will be a
minimum.
00:14:07.000 --> 00:14:10.000
Well, if you have a difference
of two squares that typically
00:14:10.000 --> 00:14:13.000
you'll have a saddle point
because depending on whether one
00:14:13.000 --> 00:14:18.000
or the other is larger,
you will have a positive or a
00:14:18.000 --> 00:14:24.000
negative quantity.
OK, so I claim there's various
00:14:24.000 --> 00:14:32.000
cases to look at.
So, let's see.
00:14:32.000 --> 00:14:34.000
So, in fact,
I claim there will be three
00:14:34.000 --> 00:14:37.000
cases.
And, that's good news for us
00:14:37.000 --> 00:14:40.000
because after all,
we want to distinguish between
00:14:40.000 --> 00:14:45.000
three possibilities.
So, let's first do away with
00:14:45.000 --> 00:14:52.000
the most complicated one.
What if 4ac minus b^2 is
00:14:52.000 --> 00:14:56.000
negative?
Well, if it's negative,
00:14:56.000 --> 00:15:00.000
then it means what I have
between the brackets is,
00:15:00.000 --> 00:15:06.000
so the first guy is obviously a
positive quantity,
00:15:06.000 --> 00:15:10.000
while the second one will be
something negative times y2.
00:15:10.000 --> 00:15:13.000
So, it will be a negative
quantity.
00:15:13.000 --> 00:15:23.000
OK, so one term is positive.
The other is negative.
00:15:23.000 --> 00:15:31.000
That tells us we actually have
a saddle point.
00:15:31.000 --> 00:15:35.000
We have, in fact,
written our function as a
00:15:35.000 --> 00:15:40.000
difference of two squares.
OK, is that convincing?
00:15:40.000 --> 00:15:42.000
So, if you want,
what I could do is actually I
00:15:42.000 --> 00:15:47.000
could change my coordinates,
have new coordinates called u
00:15:47.000 --> 00:15:50.000
equals x b over 2ay,
and v, actually,
00:15:50.000 --> 00:15:55.000
well, I could keep y,
and that it would look like the
00:15:55.000 --> 00:16:02.000
difference of squares directly.
OK, so that's the first case.
00:16:02.000 --> 00:16:12.000
The second case is where
4ac-b^2 = 0.
00:16:12.000 --> 00:16:18.000
Well, what happens if that's
zero?
00:16:18.000 --> 00:16:21.000
Then it means that this term
over there goes away.
00:16:21.000 --> 00:16:25.000
So, what we have is just one
square.
00:16:25.000 --> 00:16:29.000
OK, so what that means is
actually that our function
00:16:29.000 --> 00:16:32.000
depends only on one direction of
things.
00:16:32.000 --> 00:16:36.000
In the other direction,
it's going to actually be
00:16:36.000 --> 00:16:38.000
degenerate.
So, for example,
00:16:38.000 --> 00:16:40.000
forget all the clutter in
there.
00:16:40.000 --> 00:16:45.000
Say I give you just the
function of two variables,
00:16:45.000 --> 00:16:49.000
w equals just x^2.
So, that means it doesn't
00:16:49.000 --> 00:16:53.000
depend on y at all.
And, if I try to plot the
00:16:53.000 --> 00:16:58.000
graph, it will look like,
well, x is here.
00:16:58.000 --> 00:17:04.000
So, it will depend on x in that
way, but it doesn't depend on y
00:17:04.000 --> 00:17:10.000
at all.
So, what the graph looks like
00:17:10.000 --> 00:17:18.000
is something like that.
OK, basically it's a valley
00:17:18.000 --> 00:17:22.000
whose bottom is completely flat.
So, that means,
00:17:22.000 --> 00:17:24.000
actually, we have a degenerate
critical point.
00:17:24.000 --> 00:17:28.000
It's called degenerate because
there is a direction in which
00:17:28.000 --> 00:17:30.000
nothing happens.
And, in fact,
00:17:30.000 --> 00:17:38.000
you have critical points
everywhere along the y axis.
00:17:38.000 --> 00:17:42.000
Now, whether the square that we
have is x or something else,
00:17:42.000 --> 00:17:46.000
namely, x plus b over 2a y,
it doesn't matter.
00:17:46.000 --> 00:17:48.000
I mean, it will still get this
degenerate behavior.
00:17:48.000 --> 00:17:56.000
But there's a direction in
which nothing happens because we
00:17:56.000 --> 00:18:02.000
just have the square of one
quantity.
00:18:02.000 --> 00:18:06.000
I'm sure that 300 students
means 300 different ring tones,
00:18:06.000 --> 00:18:09.000
but I'm not eager to hear all
of them, thanks.
00:18:09.000 --> 00:18:18.000
[LAUGHTER]
OK, so, this is what's called a
00:18:18.000 --> 00:18:28.000
degenerate critical point,
and [LAUGHTER].
00:18:28.000 --> 00:18:33.000
OK, so basically we'll leave it
here.
00:18:33.000 --> 00:18:38.000
We won't actually try to figure
out further what happens,
00:18:38.000 --> 00:18:42.000
and the reason for that is that
when you have an actual
00:18:42.000 --> 00:18:44.000
function,
a general function,
00:18:44.000 --> 00:18:46.000
not just one that's quadratic
like this,
00:18:46.000 --> 00:18:50.000
then there will actually be
other terms maybe involving
00:18:50.000 --> 00:18:54.000
higher powers,
maybe x^3 or y^3 or things like
00:18:54.000 --> 00:18:56.000
that.
And then, they will mess up
00:18:56.000 --> 00:19:00.000
what happens in this valley.
And, it's a situation where we
00:19:00.000 --> 00:19:03.000
won't be able,
actually, to tell automatically
00:19:03.000 --> 00:19:06.000
just by looking at second
derivatives what happens.
00:19:06.000 --> 00:19:09.000
See, for example,
in a function of one variable,
00:19:09.000 --> 00:19:12.000
if you have just a function of
one variable,
00:19:12.000 --> 00:19:14.000
say, f of x equals x to the
five,
00:19:14.000 --> 00:19:18.000
well, if you try to decide what
type of point the origin is,
00:19:18.000 --> 00:19:20.000
you're going to take the second
derivative.
00:19:20.000 --> 00:19:23.000
It will be zero,
and then you can conclude.
00:19:23.000 --> 00:19:26.000
Those things depend on higher
order derivatives.
00:19:26.000 --> 00:19:29.000
So, we just won't like that
case.
00:19:29.000 --> 00:19:34.000
We just won't try to figure out
what's going on here.
00:19:34.000 --> 00:19:40.000
Now, the last situation is if
4ac-b^2 is positive.
00:19:40.000 --> 00:19:45.000
So, then, that means that
actually we've written things.
00:19:45.000 --> 00:19:52.000
The big bracket up there is a
sum of two squares.
00:19:52.000 --> 00:20:00.000
So, that means that we've
written w as one over 4a times
00:20:00.000 --> 00:20:08.000
plus something squared plus
something else squared,
00:20:08.000 --> 00:20:12.000
OK?
So, these guys have the same
00:20:12.000 --> 00:20:18.000
sign, and that means that this
term here will always be greater
00:20:18.000 --> 00:20:22.000
than or equal to zero.
And that means that we should
00:20:22.000 --> 00:20:24.000
either have a maximum or
minimum.
00:20:24.000 --> 00:20:29.000
How we find out which one it is?
Well, we look at the sign of a,
00:20:29.000 --> 00:20:30.000
exactly.
OK?
00:20:30.000 --> 00:20:35.000
So, there's two sub-cases.
One is if a is positive,
00:20:35.000 --> 00:20:40.000
then, this quantity overall
will always be nonnegative.
00:20:40.000 --> 00:20:54.000
And that means we have a
minimum, OK?
00:20:54.000 --> 00:20:58.000
And, if a is negative on the
other hand,
00:20:58.000 --> 00:21:01.000
so that means that we multiply
this positive quantity by a
00:21:01.000 --> 00:21:04.000
negative number,
we get something that's always
00:21:04.000 --> 00:21:10.000
negative.
So, zero is actually the
00:21:10.000 --> 00:21:18.000
maximum.
OK, is that clear for everyone?
00:21:18.000 --> 00:21:29.000
Yes?
Sorry, yeah,
00:21:29.000 --> 00:21:34.000
so I said in the example w
equals x^2, it doesn't depend on
00:21:34.000 --> 00:21:37.000
y.
So, the more general situation
00:21:37.000 --> 00:21:44.000
is w equals some constant.
Well, I guess it's a times (x b
00:21:44.000 --> 00:21:48.000
over 2a times y)^2.
So, it does depend on x and y,
00:21:48.000 --> 00:21:51.000
but it only depends on this
combination.
00:21:51.000 --> 00:21:54.000
OK, so if I choose to move in
some other perpendicular
00:21:54.000 --> 00:21:58.000
direction,
in the direction where this
00:21:58.000 --> 00:22:02.000
remains constant,
so maybe if I set x equals
00:22:02.000 --> 00:22:06.000
minus b over 2a y,
then this remains zero all the
00:22:06.000 --> 00:22:08.000
time.
So, there's a degenerate
00:22:08.000 --> 00:22:11.000
direction in which I stay at the
minimum or maximum,
00:22:11.000 --> 00:22:15.000
or whatever it is that I have.
OK, so that's why it's called
00:22:15.000 --> 00:22:17.000
degenerate.
There is a direction in which
00:22:17.000 --> 00:22:29.000
nothing happens.
OK, yes?
00:22:29.000 --> 00:22:31.000
Yes, yeah, so that's a very
good question.
00:22:31.000 --> 00:22:33.000
So, there's going to be the
second derivative test.
00:22:33.000 --> 00:22:36.000
Why do not have derivatives yet?
Well, that's because I've been
00:22:36.000 --> 00:22:39.000
looking at this special example
where we have a function like
00:22:39.000 --> 00:22:41.000
this.
And, so I don't actually need
00:22:41.000 --> 00:22:43.000
to take derivatives yet.
But, secretly,
00:22:43.000 --> 00:22:46.000
that's because a,
b, and c will be the second
00:22:46.000 --> 00:22:49.000
derivatives of the function,
actually, 2a,
00:22:49.000 --> 00:22:52.000
b, and 2c.
So now, we are going to go to
00:22:52.000 --> 00:22:54.000
general function.
And there, instead of having
00:22:54.000 --> 00:22:57.000
these coefficients a,
b, and c given to us,
00:22:57.000 --> 00:23:00.000
we'll have to compute them as
second derivatives.
00:23:00.000 --> 00:23:03.000
OK, so here,
I'm basically setting the stage
00:23:03.000 --> 00:23:07.000
for what will be the actual
criterion we'll use using second
00:23:07.000 --> 00:23:13.000
derivatives.
Yes?
00:23:13.000 --> 00:23:16.000
So, yeah, so what you have a
degenerate critical point,
00:23:16.000 --> 00:23:20.000
it could be a degenerate
minimum, or a degenerate maximum
00:23:20.000 --> 00:23:23.000
depending on the sign of a.
But, in general,
00:23:23.000 --> 00:23:26.000
once you start having
functions, you don't really know
00:23:26.000 --> 00:23:30.000
what will happen anymore.
It could also be a degenerate
00:23:30.000 --> 00:23:36.000
saddle, and so on.
So, we won't really be able to
00:23:36.000 --> 00:23:40.000
tell.
Yes?
00:23:40.000 --> 00:23:43.000
It is possible to have a
degenerate saddle point.
00:23:43.000 --> 00:23:46.000
For example,
if I gave you x^3 y^3,
00:23:46.000 --> 00:23:49.000
you can convince yourself that
if you take x and y to be
00:23:49.000 --> 00:23:53.000
negative, it will be negative.
If x and y are positive,
00:23:53.000 --> 00:23:55.000
it's positive.
And, it has a very degenerate
00:23:55.000 --> 00:23:59.000
critical point at the origin.
So, that's a degenerate saddle
00:23:59.000 --> 00:24:01.000
point.
We don't see it here because
00:24:01.000 --> 00:24:04.000
that doesn't happen if you have
only quadratic terms like that.
00:24:04.000 --> 00:24:12.000
You need to have higher-order
terms to see it happen.
00:24:12.000 --> 00:24:23.000
OK.
OK, so let's continue.
00:24:23.000 --> 00:24:27.000
Before we continue,
but see, I wanted to point out
00:24:27.000 --> 00:24:30.000
one small thing.
So, here, we have the magic
00:24:30.000 --> 00:24:34.000
quantity, 4ac minus b^2.
You've probably seen that
00:24:34.000 --> 00:24:37.000
before in your life.
Yet, it looks like the
00:24:37.000 --> 00:24:40.000
quadratic formula,
except that one involves
00:24:40.000 --> 00:24:43.000
b^2-4ac.
But that's really the same
00:24:43.000 --> 00:24:47.000
thing.
OK, so let's see,
00:24:47.000 --> 00:24:57.000
where does the quadratic
formula come in here?
00:24:57.000 --> 00:25:00.000
Well, let me write things
differently.
00:25:00.000 --> 00:25:03.000
OK, so we've manipulated
things, and got into a
00:25:03.000 --> 00:25:08.000
conclusion.
But, let me just do a different
00:25:08.000 --> 00:25:14.000
manipulation,
and write this now instead as
00:25:14.000 --> 00:25:23.000
y^2 times a times x over y
squared plus b(x over y) plus c.
00:25:23.000 --> 00:25:28.000
OK, see, that's the same thing
that I had before.
00:25:28.000 --> 00:25:35.000
Well, so now this quantity here
is always nonnegative.
00:25:35.000 --> 00:25:39.000
What about this one?
Well, of course,
00:25:39.000 --> 00:25:43.000
this one depends on x over y.
It means it depends on which
00:25:43.000 --> 00:25:45.000
direction you're going to move
away from the origin,
00:25:45.000 --> 00:25:48.000
which ratio between x and y you
will consider.
00:25:48.000 --> 00:25:51.000
But, I claim there's two
situations.
00:25:51.000 --> 00:25:57.000
One is, so, let's try to
reformulate things.
00:25:57.000 --> 00:26:04.000
So, if a discriminate here is
positive, then it means that
00:26:04.000 --> 00:26:10.000
these have roots and these have
solutions.
00:26:10.000 --> 00:26:19.000
And, that means that this
quantity can be both positive
00:26:19.000 --> 00:26:24.000
and negative.
This quantity takes positive
00:26:24.000 --> 00:26:31.000
and negative values.
One way to convince yourself is
00:26:31.000 --> 00:26:37.000
just to, you know,
plot at^2 bt c.
00:26:37.000 --> 00:26:43.000
You know that there's two roots.
So, it might look like this,
00:26:43.000 --> 00:26:48.000
or might look like that
depending on the sign of a.
00:26:48.000 --> 00:26:52.000
But, in either case,
it will take values of both
00:26:52.000 --> 00:26:54.000
signs.
So, that means that your
00:26:54.000 --> 00:26:56.000
function will take values of
both signs.
00:27:04.000 --> 00:27:13.000
The value takes both positive
and negative values.
00:27:13.000 --> 00:27:21.000
And, so that means we have a
saddle point,
00:27:21.000 --> 00:27:28.000
while the other situation,
when b^2-4ac is negative -- --
00:27:28.000 --> 00:27:36.000
means that this equation is
quadratic never takes the value,
00:27:36.000 --> 00:27:39.000
zero.
So, it's always positive or
00:27:39.000 --> 00:27:42.000
it's always negative,
depending on the sign of a.
00:27:42.000 --> 00:27:48.000
So, the other case is if
b^2-4ac is negative,
00:27:48.000 --> 00:27:53.000
then the quadratic doesn't have
a solution.
00:27:53.000 --> 00:27:58.000
And it could look like this or
like that depending on whether a
00:27:58.000 --> 00:28:03.000
is positive or a is negative.
So, in particular,
00:28:03.000 --> 00:28:12.000
that means that ax over y2 plus
bx over y plus c is always
00:28:12.000 --> 00:28:21.000
positive or always negative
depending on the sign of a.
00:28:21.000 --> 00:28:23.000
And then, that tells us that
our function,
00:28:23.000 --> 00:28:25.000
w, will be always positive or
always negative.
00:28:25.000 --> 00:28:28.000
And then we'll get a minimum or
maximum.
00:28:40.000 --> 00:28:44.000
OK, we'll have a min or a max
depending on which situation we
00:28:44.000 --> 00:28:47.000
are in.
OK, so that's another way to
00:28:47.000 --> 00:28:51.000
derive the same answer.
And now, you see here why the
00:28:51.000 --> 00:28:55.000
discriminate plays a role.
That's because it exactly tells
00:28:55.000 --> 00:28:59.000
you whether this quadratic
quantity has always the same
00:28:59.000 --> 00:29:04.000
sign,
or whether it can actually
00:29:04.000 --> 00:29:12.000
cross the value,
zero, when you have the root of
00:29:12.000 --> 00:29:16.000
a quadratic.
OK, so hopefully at this stage
00:29:16.000 --> 00:29:20.000
you are happy with one of the
two explanations,
00:29:20.000 --> 00:29:23.000
at least.
And now, you are willing to
00:29:23.000 --> 00:29:26.000
believe, I hope,
that we have basically a way of
00:29:26.000 --> 00:29:30.000
deciding what type of critical
point we have in the special
00:29:30.000 --> 00:29:32.000
case of a quadratic function.
00:29:58.000 --> 00:30:05.000
OK, so, now what do we do with
the general function?
00:30:05.000 --> 00:30:19.000
Well, so in general,
we want to look at second
00:30:19.000 --> 00:30:24.000
derivatives.
OK, so now we are getting to
00:30:24.000 --> 00:30:27.000
the real stuff.
So, how many second derivatives
00:30:27.000 --> 00:30:29.000
do we have?
That's maybe the first thing we
00:30:29.000 --> 00:30:32.000
should figure out.
Well, we can take the
00:30:32.000 --> 00:30:39.000
derivative first with respect to
x, and then again with respect
00:30:39.000 --> 00:30:44.000
to x.
OK, that gives us something we
00:30:44.000 --> 00:30:54.000
denote by partial square f over
partial x squared or fxx.
00:30:54.000 --> 00:31:00.000
Then, there's another one which
is fxy, which means you take the
00:31:00.000 --> 00:31:05.000
derivative with respect to x,
and then with respect to y.
00:31:05.000 --> 00:31:09.000
Another thing you can do,
is do first derivative respect
00:31:09.000 --> 00:31:12.000
to y, and then with respect to
x.
00:31:12.000 --> 00:31:17.000
That would be fyx.
Well, good news.
00:31:17.000 --> 00:31:22.000
These are actually always equal
to each other.
00:31:22.000 --> 00:31:26.000
OK, so it's the fact that we
will admit, it's actually not
00:31:26.000 --> 00:31:30.000
very hard to check.
So these are always the same.
00:31:30.000 --> 00:31:33.000
We don't need to worry about
which one we do.
00:31:33.000 --> 00:31:36.000
That's one computation that we
won't need to do.
00:31:36.000 --> 00:31:43.000
We can save a bit of effort.
And then, we have the last one,
00:31:43.000 --> 00:31:51.000
namely, the second partial with
respect to y and y fyy.
00:31:51.000 --> 00:32:00.000
OK, so we have three of them.
So, what does the second
00:32:00.000 --> 00:32:02.000
derivative test say?
00:32:16.000 --> 00:32:22.000
It says, say that you have a
critical point (x0,
00:32:22.000 --> 00:32:27.000
y0) of a function of two
variables, f,
00:32:27.000 --> 00:32:34.000
and then let's compute the
partial derivatives.
00:32:34.000 --> 00:32:41.000
So, let's call capital A the
second derivative with respect
00:32:41.000 --> 00:32:45.000
to x.
Let's call capital B the second
00:32:45.000 --> 00:32:49.000
derivative with respect to x and
y.
00:32:49.000 --> 00:32:55.000
And C equals fyy at this point,
OK?
00:32:55.000 --> 00:32:59.000
So, these are just numbers
because we first compute the
00:32:59.000 --> 00:33:02.000
second derivative,
and then we plug in the values
00:33:02.000 --> 00:33:04.000
of x and y at the critical
point.
00:33:04.000 --> 00:33:14.000
So, these will just be numbers.
And now, what we do is we look
00:33:14.000 --> 00:33:21.000
at the quantity AC-B^2.
I am not forgetting the four.
00:33:21.000 --> 00:33:26.000
You will see why there isn't
one.
00:33:26.000 --> 00:33:31.000
So, if AC-B^2 is positive,
then there's two sub-cases.
00:33:31.000 --> 00:33:39.000
If A is positive,
then it's local minimum.
00:33:50.000 --> 00:33:56.000
The second case,
so, still, if AC-B^2 is
00:33:56.000 --> 00:34:04.000
positive, but A is negative,
then it's going to be a local
00:34:04.000 --> 00:34:11.000
maximum.
And, if AC-B^2 is negative,
00:34:11.000 --> 00:34:17.000
then it's a saddle point,
and finally,
00:34:17.000 --> 00:34:24.000
if AC-B^2 is zero,
then we actually cannot
00:34:24.000 --> 00:34:28.000
compute.
We don't know whether it's
00:34:28.000 --> 00:34:33.000
going to be a minimum,
a maximum, or a saddle.
00:34:33.000 --> 00:34:37.000
We know it's degenerate in some
way, but we don't know what type
00:34:37.000 --> 00:34:40.000
of point it is.
OK, so that's actually what you
00:34:40.000 --> 00:34:43.000
need to remember.
If you are formula oriented,
00:34:43.000 --> 00:34:45.000
that's all you need to remember
about today.
00:34:45.000 --> 00:34:53.000
But, let's try to understand
why, how this comes out of what
00:34:53.000 --> 00:34:59.000
we had there.
OK, so, I think maybe I
00:34:59.000 --> 00:35:05.000
actually want to keep,
so maybe I want to keep this
00:35:05.000 --> 00:35:06.000
middle board because it actually
has,
00:35:06.000 --> 00:35:09.000
you know, the recipe that we
found before the quadratic
00:35:09.000 --> 00:35:12.000
function.
So, let me move directly over
00:35:12.000 --> 00:35:16.000
there and try to relate our old
recipe with the new.
00:35:43.000 --> 00:35:50.000
OK, you are easily amused.
OK, so first,
00:35:50.000 --> 00:35:57.000
let's check that these two
things say the same thing in the
00:35:57.000 --> 00:36:01.000
special case that we are looking
at.
00:36:01.000 --> 00:36:12.000
OK, so let's verify in the
special case where the function
00:36:12.000 --> 00:36:22.000
was ax^2 bxy cy^2.
So -- Well, what is the second
00:36:22.000 --> 00:36:28.000
derivative with respect to x and
x?
00:36:28.000 --> 00:36:31.000
If I take the second derivative
with respect to x and x,
00:36:31.000 --> 00:36:34.000
so first I want to take maybe
the derivative with respect to
00:36:34.000 --> 00:36:37.000
x.
But first, let's take the first
00:36:37.000 --> 00:36:46.000
partial, Wx.
That will be 2ax by, right?
00:36:46.000 --> 00:36:50.000
So, Wxx will be,
well, let's take a partial with
00:36:50.000 --> 00:36:54.000
respect to x again.
That's 2a.
00:36:54.000 --> 00:37:02.000
Wxy, I take the partial respect
to y, and we'll get b.
00:37:02.000 --> 00:37:06.000
OK, now we need,
also, the partial with respect
00:37:06.000 --> 00:37:13.000
to y.
So, Wy is bx 2cy.
00:37:13.000 --> 00:37:17.000
In case you don't believe what
I told you about the mixed
00:37:17.000 --> 00:37:21.000
partials, Wyx,
well, you can check.
00:37:21.000 --> 00:37:24.000
And it's, again, b.
So, they are,
00:37:24.000 --> 00:37:30.000
indeed, the same thing.
And, Wyy will be 2c.
00:37:30.000 --> 00:37:39.000
So, if we now look at these
quantities, that tells us,
00:37:39.000 --> 00:37:46.000
well, big A is two little a,
big B is little b,
00:37:46.000 --> 00:37:55.000
big C is two little c.
So, AC-B^2 is what we used to
00:37:55.000 --> 00:38:04.000
call four little ac minus b2.
OK, ooh.
00:38:04.000 --> 00:38:07.000
[LAUGHTER]
So, now you can compare the
00:38:07.000 --> 00:38:10.000
cases.
They are not listed in the same
00:38:10.000 --> 00:38:14.000
order just to make it harder.
So, we said first,
00:38:14.000 --> 00:38:20.000
so the saddle case is when
AC-B^2 in big letters is
00:38:20.000 --> 00:38:26.000
negative, that's the same as
4ac-b2 in lower case is
00:38:26.000 --> 00:38:30.000
negative.
The case where capital AC-B2 is
00:38:30.000 --> 00:38:35.000
positive, local min and local
max corresponds to this one.
00:38:35.000 --> 00:38:40.000
And, the case where we can't
conclude was what used to be the
00:38:40.000 --> 00:38:44.000
degenerate one.
OK, so at least we don't seem
00:38:44.000 --> 00:38:48.000
to have messed up when copying
the formula.
00:38:48.000 --> 00:38:56.000
Now, why does that work more
generally than that?
00:38:56.000 --> 00:39:03.000
Well, the answer that is,
again, Taylor approximation.
00:39:03.000 --> 00:39:16.000
Aww.
OK, so let me just do here
00:39:16.000 --> 00:39:22.000
quadratic approximation.
So, quadratic approximation
00:39:22.000 --> 00:39:25.000
tells me the following thing.
It tells me,
00:39:25.000 --> 00:39:30.000
if I have a function,
f of xy, and I want to
00:39:30.000 --> 00:39:37.000
understand the change in f when
I change x and y a little bit.
00:39:37.000 --> 00:39:40.000
Well, there's the first-order
terms.
00:39:40.000 --> 00:39:43.000
There is the linear terms that
by now you should know and be
00:39:43.000 --> 00:39:51.000
comfortable with.
That's fx times the change in x.
00:39:51.000 --> 00:39:56.000
And then, there's fy times the
change in y.
00:39:56.000 --> 00:40:00.000
OK, that's the starting point.
But now, of course,
00:40:00.000 --> 00:40:03.000
if x and y, sorry,
if we are at the critical
00:40:03.000 --> 00:40:09.000
point, then that's going to be
zero at the critical point.
00:40:09.000 --> 00:40:16.000
So, that term actually goes
away, and that's also zero at
00:40:16.000 --> 00:40:22.000
the critical point.
So, that term also goes away.
00:40:22.000 --> 00:40:24.000
OK, so linear approximation is
really no good.
00:40:24.000 --> 00:40:27.000
We need more terms.
So, what are the next terms?
00:40:27.000 --> 00:40:35.000
Well, the next terms are
quadratic terms,
00:40:35.000 --> 00:40:38.000
and so I mean,
if you remember the Taylor
00:40:38.000 --> 00:40:42.000
formula for a function of a
single variable,
00:40:42.000 --> 00:40:46.000
there was the derivative times
x minus x0 plus one half of a
00:40:46.000 --> 00:40:51.000
second derivative times x-x0^2.
And see, this side here is
00:40:51.000 --> 00:40:55.000
really Taylor approximation in
one variable looking only at x.
00:40:55.000 --> 00:40:57.000
But of course,
we also have terms involving y,
00:40:57.000 --> 00:41:00.000
and terms involving
simultaneously x and y.
00:41:00.000 --> 00:41:10.000
And, these terms are fxy times
change in x times change in y
00:41:10.000 --> 00:41:17.000
plus one half of fyy(y-y0)^2.
There's no one half in the
00:41:17.000 --> 00:41:20.000
middle because,
in fact, you would have two
00:41:20.000 --> 00:41:24.000
terms, one for xy,
one for yx, but they are the
00:41:24.000 --> 00:41:26.000
same.
And then, if you want to
00:41:26.000 --> 00:41:29.000
continue, there is actually
cubic terms involving the third
00:41:29.000 --> 00:41:32.000
derivatives, and so on,
but we are not actually looking
00:41:32.000 --> 00:41:34.000
at them.
And so, now,
00:41:34.000 --> 00:41:39.000
when we do this approximation,
well, the type of critical
00:41:39.000 --> 00:41:45.000
point remains the same when we
replace the function by this
00:41:45.000 --> 00:41:48.000
approximation.
And so, we can apply the
00:41:48.000 --> 00:41:53.000
argument that we used to deduce
things in the quadratic case.
00:41:53.000 --> 00:41:55.000
In fact, it still works in the
general case using this
00:41:55.000 --> 00:41:57.000
approximation formula.
00:42:12.000 --> 00:42:26.000
So -- The general case reduces
to the quadratic case.
00:42:26.000 --> 00:42:31.000
And now, you see actually why,
well, here you see,
00:42:31.000 --> 00:42:36.000
again, how this coefficient
which we used to call little a
00:42:36.000 --> 00:42:41.000
is also one half of capital A.
And same here:
00:42:41.000 --> 00:42:47.000
this coefficient is what we
call capital B or little b,
00:42:47.000 --> 00:42:52.000
and this coefficient here is
what we called little c or one
00:42:52.000 --> 00:42:57.000
half of capital C.
And then, when you replace
00:42:57.000 --> 00:43:02.000
these into the various cases
that we had here,
00:43:02.000 --> 00:43:06.000
you end up with the second
derivative test.
00:43:06.000 --> 00:43:08.000
So, what about the degenerate
case?
00:43:08.000 --> 00:43:11.000
Why can't we just say,
well, it's going to be a
00:43:11.000 --> 00:43:16.000
degenerate critical point?
So, the reason is that this
00:43:16.000 --> 00:43:20.000
approximation formula is
reasonable only if the higher
00:43:20.000 --> 00:43:24.000
order terms are negligible.
OK, so in fact,
00:43:24.000 --> 00:43:27.000
secretly, there's more terms.
This is only an approximation.
00:43:27.000 --> 00:43:30.000
There would be terms involving
third derivatives,
00:43:30.000 --> 00:43:34.000
and maybe even beyond that.
And, so it is not to generate
00:43:34.000 --> 00:43:37.000
case,
they don't actually matter
00:43:37.000 --> 00:43:39.000
because the shape of the
function,
00:43:39.000 --> 00:43:42.000
the shape of the graph,
is actually determined by the
00:43:42.000 --> 00:43:45.000
quadratic terms.
But, in the degenerate case,
00:43:45.000 --> 00:43:49.000
see, if I start with this and I
add something even very,
00:43:49.000 --> 00:43:53.000
very small along the y axis,
then that can be enough to bend
00:43:53.000 --> 00:43:56.000
this very slightly up or
slightly down,
00:43:56.000 --> 00:44:00.000
and turn my degenerate point in
to either a minimum or a saddle
00:44:00.000 --> 00:44:03.000
point.
And, I won't be able to tell
00:44:03.000 --> 00:44:06.000
until I go further in the list
of derivatives.
00:44:06.000 --> 00:44:14.000
So, in the degenerate case,
what actually happens depends
00:44:14.000 --> 00:44:20.000
on the higher order derivatives.
00:44:38.000 --> 00:44:42.000
So, we will need to analyze
things more carefully.
00:44:42.000 --> 00:44:45.000
Well, we're not going to bother
with that in this class.
00:44:45.000 --> 00:44:52.000
So, we'll just say,
well, we cannot compute,
00:44:52.000 --> 00:44:54.000
OK?
I mean, you have to realize
00:44:54.000 --> 00:44:57.000
that in real life,
you have to be extremely
00:44:57.000 --> 00:45:02.000
unlucky for this quantity to end
up being exactly 0.
00:45:02.000 --> 00:45:03.000
[LAUGHTER]
Well, if that happens,
00:45:03.000 --> 00:45:05.000
then what you should do is
maybe try by inspection.
00:45:05.000 --> 00:45:08.000
See if there's a good reason
why the function should always
00:45:08.000 --> 00:45:11.000
be positive or always be
negative, or something.
00:45:11.000 --> 00:45:16.000
Or, you know,
plot it on a computer and see
00:45:16.000 --> 00:45:23.000
what happens.
But, otherwise we can't compute.
00:45:23.000 --> 00:45:33.000
OK, so let's do an example.
So, probably I should leave
00:45:33.000 --> 00:45:39.000
this on so that we still have
the test with us.
00:45:39.000 --> 00:45:42.000
And, instead,
OK, so I'll do my example here.
00:46:20.000 --> 00:46:30.000
OK, so just an example.
Let's look at f of (x,
00:46:30.000 --> 00:46:37.000
y) = x y 1/xy,
where x and y are positive.
00:46:37.000 --> 00:46:39.000
So, I'm looking only at the
first quadrant.
00:46:39.000 --> 00:46:42.000
OK, I mean, I'm doing this
because I don't want the
00:46:42.000 --> 00:46:46.000
denominator to become zero.
So, I'm just looking at the
00:46:46.000 --> 00:46:50.000
situation.
So, let's look first for,
00:46:50.000 --> 00:46:55.000
so, the question will be,
what are the minimum and
00:46:55.000 --> 00:47:03.000
maximum of this function?
So, the first thing we should
00:47:03.000 --> 00:47:12.000
do to answer this question is
look for critical points,
00:47:12.000 --> 00:47:15.000
OK?
So, for that,
00:47:15.000 --> 00:47:19.000
we have to compute the first
derivatives.
00:47:19.000 --> 00:47:34.000
OK, so fx is one minus one over
x^2y, OK?
00:47:34.000 --> 00:47:39.000
Take the derivative of one over
x, that's negative one over x^2.
00:47:39.000 --> 00:47:44.000
And, we'll want to set that
equal to zero.
00:47:44.000 --> 00:47:50.000
And fy is one minus one over
xy^2.
00:47:50.000 --> 00:47:54.000
And, we want to set that equal
to zero.
00:47:54.000 --> 00:47:59.000
So, what are the equations we
have to solve?
00:47:59.000 --> 00:48:05.000
Well, I guess x^2y equals one,
I mean, if I move this guy over
00:48:05.000 --> 00:48:09.000
here I get one over x^2y equals
one.
00:48:09.000 --> 00:48:14.000
That's x^2y equals one,
and xy^2 equals one.
00:48:14.000 --> 00:48:18.000
What do you get by comparing
these two?
00:48:18.000 --> 00:48:21.000
Well, x and y should both be,
OK, so yeah,
00:48:21.000 --> 00:48:24.000
I agree with you that one and
one is a solution.
00:48:24.000 --> 00:48:27.000
Why is it the only one?
So, first, if I divide this one
00:48:27.000 --> 00:48:29.000
by that one, I get x over y
equals one.
00:48:29.000 --> 00:48:34.000
So, it tells me x equals y.
And then, if x equals y,
00:48:34.000 --> 00:48:40.000
then if I put that into here,
it will give me y^3 equals one,
00:48:40.000 --> 00:48:44.000
which tells me y equals one,
and therefore,
00:48:44.000 --> 00:48:50.000
x equals one as well.
OK, so, there's only one
00:48:50.000 --> 00:48:54.000
solution.
There's only one critical
00:48:54.000 --> 00:48:58.000
point, which is going to be
(1,1).
00:48:58.000 --> 00:49:09.000
OK, so, now here's where you do
a bit of work.
00:49:09.000 --> 00:49:18.000
What do you think of that
critical point?
00:49:18.000 --> 00:49:25.000
OK, I see some valid votes.
I see some, OK,
00:49:25.000 --> 00:49:28.000
I see a lot of people answering
four.
00:49:28.000 --> 00:49:30.000
[LAUGHTER]
that seems to suggest that
00:49:30.000 --> 00:49:34.000
maybe you haven't completed the
second derivative yet.
00:49:34.000 --> 00:49:37.000
Yes, I see someone giving the
correct answer.
00:49:37.000 --> 00:49:41.000
I see some people not giving
quite the correct answer.
00:49:41.000 --> 00:49:43.000
I see more and more correct
answers.
00:49:43.000 --> 00:49:49.000
OK, so let's see.
To figure out what type of
00:49:49.000 --> 00:49:52.000
point is, we should compute the
second partial derivatives.
00:49:52.000 --> 00:50:02.000
So, fxx is, what do we get what
we take the derivative of this
00:50:02.000 --> 00:50:11.000
with respect to x?
Two over x^3y, OK?
00:50:11.000 --> 00:50:25.000
So, at our point, a will be 2.
Fxy will be one over x^2y^2.
00:50:25.000 --> 00:50:37.000
So, B will be one.
And, Fyy is going to be two
00:50:37.000 --> 00:50:42.000
over xy^3.
So, C will be two.
00:50:42.000 --> 00:50:51.000
And so that tells us,
well, AC-B^2 is four minus one.
00:50:51.000 --> 00:51:02.000
Sorry, I should probably use a
different blackboard for that.
00:51:02.000 --> 00:51:06.000
AC-B2 is two times two minus
1^2 is three.
00:51:06.000 --> 00:51:10.000
It's positive.
That tells us we are either a
00:51:10.000 --> 00:51:17.000
local minimum or local maximum.
And, A is positive.
00:51:17.000 --> 00:51:21.000
So, it's a local minimum.
And, in fact,
00:51:21.000 --> 00:51:23.000
you can check it's the global
minimum.
00:51:23.000 --> 00:51:29.000
What about the maximum?
Well, if a maximum is not
00:51:29.000 --> 00:51:32.000
actually at a critical point,
it's on the boundary,
00:51:32.000 --> 00:51:35.000
or at infinity.
See, so we have actually to
00:51:35.000 --> 00:51:39.000
check what happens when x and y
go to zero or to infinity.
00:51:39.000 --> 00:51:42.000
Well, if that happens,
if x or y goes to infinity,
00:51:42.000 --> 00:51:44.000
then the function goes to
infinity.
00:51:44.000 --> 00:51:48.000
Also, if x or y goes to zero,
then one over xy goes to
00:51:48.000 --> 00:51:51.000
infinity.
So, the maximum,
00:51:51.000 --> 00:51:59.000
well, the function goes to
infinity when x goes to infinity
00:51:59.000 --> 00:52:05.000
or y goes to infinity,
or x and y go to zero.
00:52:05.000 --> 00:52:07.000
So, it's not at a critical
point.
00:52:07.000 --> 00:52:10.000
OK, so, in general,
we have to check both the
00:52:10.000 --> 00:52:13.000
critical points and the
boundaries to decide what
00:52:13.000 --> 00:52:15.000
happens.
OK, the end.
00:52:15.000 --> 00:52:18.000
Have a nice weekend.