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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

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decide whether a typical point
is a minimum,

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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,

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we looked at critical points of
functions of several variables.

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And, so a critical point
functions, we have two values,

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x and y.
That's a point where the

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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.

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00:01:20,000 --> 00:01:24,000
So, maybe I should show the
function on this contour

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plot,there is local maxima,
which are like that.

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And, there's saddle points
which are neither minima nor

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maxima.
And, of course,

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00:01:37,000 --> 00:01:41,000
if you have a real function,
then it would be more

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complicated.
It will have several critical

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points.
So, this example here,

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well, you see on the plot that
there is two maxima.

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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,

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you can see them on the contour
plot.

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00:02:02,000 --> 00:02:07,000
On the contour plot,
you see the maxima because the

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level curves become circles that
now down and shrink to the

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maximum.
And, you can see the saddle

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00:02:15,000 --> 00:02:18,000
point because here you have this
level curve that makes a figure

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00:02:18,000 --> 00:02:20,000
eight.
It crosses itself.

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00:02:20,000 --> 00:02:25,000
And, if you move up or down
here, so along the y direction,

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the values of the function will
decrease.

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00:02:28,000 --> 00:02:32,000
Along the x direction,
the values will increase.

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00:02:32,000 --> 00:02:37,000
So, you can see usually quite
easily where are the critical

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00:02:37,000 --> 00:02:42,000
points just by looking either at
the graph or at the contour

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00:02:42,000 --> 00:02:44,000
plots.
So, the only thing with the

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00:02:44,000 --> 00:02:47,000
contour plots is you need to
read the values to tell a

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00:02:47,000 --> 00:02:51,000
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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00:03:44,000 --> 00:03:55,000
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

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out, well,
first of all,

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to decide where the function is
the largest,

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00:04:09,000 --> 00:04:12,000
in general you'll have actually
to compare the values.

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00:04:12,000 --> 00:04:14,000
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,

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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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00:04:43,000 --> 00:04:47,000
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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00:04:49,000 --> 00:04:53,000
Well, it looks like the minimum
is actually out there on the

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00:04:53,000 --> 00:04:56,000
boundary or at infinity.
So, that's another feature.

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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

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where, actually,
the minimum and maximum will

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be.
So, in general,

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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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00:05:48,000 --> 00:05:52,000
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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00:05:55,000 --> 00:06:00,000
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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00:06:04,000 --> 00:06:09,000
the question of,
you know, what's the type of

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the critical point?
So, we'll use something that's

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known as the second derivative
test.

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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,

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00:06:29,000 --> 00:06:32,000
namely, the function of one
variable.

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00:06:32,000 --> 00:06:34,000
If the derivative is zero,
then you know that you should

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00:06:34,000 --> 00:06:38,000
look at the second derivative.
And, that will tell you whether

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00:06:38,000 --> 00:06:41,000
it's curving up or down whether
you have a local max and the

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local min.
And, the main problem here is,

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00:06:44,000 --> 00:06:46,000
of course, we have more
possible situations,

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00:06:46,000 --> 00:06:48,000
and we have several
derivatives.

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So, we have to think a bit
harder about how we'll decide.

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00:06:52,000 --> 00:06:56,000
But, it will again involve the
second derivative.

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OK, so let's start with just an
easy example that will be useful

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00:07:01,000 --> 00:07:06,000
to us because actually it will
provide the basis for the

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general method.
OK, so we are first going to

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00:07:10,000 --> 00:07:15,000
consider a case where we have a
function that's actually just

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quadratic.
So, let's say I have a

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function, W of (x,y) that's of
the form ax^2 bxy cy^2.

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00:07:28,000 --> 00:07:32,000
OK, so this guy has a critical
point at the origin because if

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you take the derivative with
respect to x,

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well, and if you plug x equals
y equals zero,

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you'll get zero,
and same with respect to y.

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00:07:42,000 --> 00:07:44,000
You can also see,
if you try to do a linear

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approximation of this,
well, all these guys are much

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smaller than x and y when x and
y are small.

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So, the linear approximation,
the tangent plane to the graph

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00:07:55,000 --> 00:07:59,000
is really just w=0.
OK, so, how do we do it?

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00:07:59,000 --> 00:08:03,000
Well, yesterday we actually did
an example.

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00:08:03,000 --> 00:08:09,000
It was a bit more complicated
than that, but let me do it,

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00:08:09,000 --> 00:08:13,000
so remember,
we were looking at something

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00:08:13,000 --> 00:08:19,000
that started with x^2 2xy 3y^2.
And, there were other terms.

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00:08:19,000 --> 00:08:23,000
But, let's forget them now.
And, what we did is we said,

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00:08:23,000 --> 00:08:28,000
well, we can rewrite this as (x
y)^2 2y^2.

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00:08:28,000 --> 00:08:31,000
And now, this is a sum of two
squares.

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00:08:31,000 --> 00:08:35,000
So, each of these guys has to
be nonnegative.

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00:08:35,000 --> 00:08:40,000
And so, the origin will be a
minimum.

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00:08:40,000 --> 00:08:44,000
Well, it turns out we can do
something similar in general no

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matter what the values of a,
b, and c are.

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We'll just try to first
complete things to a square.

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OK, so let's do that.
So, in general,

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well, let me be slightly less
general, and let me assume that

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00:09:01,000 --> 00:09:08,000
a is not zero because otherwise
I can't do what I'm going to do.

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00:09:08,000 --> 00:09:20,000
So, I'm going to write this as
a times x^2 plus b over axy.

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00:09:20,000 --> 00:09:25,000
And then I have my cy^2.
And now this looks like the

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beginning of the square of
something, OK,

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00:09:28,000 --> 00:09:31,000
just like what we did over
there.

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So, what is it the square of?
Well, you'd start with x plus I

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claim if I put b over 2a times y
and I square it,

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00:09:45,000 --> 00:09:52,000
then see the cross term two
times x times b over 2a y will

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00:09:52,000 --> 00:09:57,000
become b over axy.
Of course, now I also get some

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00:09:57,000 --> 00:10:01,000
y squares out of this.
How many y squares do I get?

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00:10:01,000 --> 00:10:05,000
Well, I get b^2 over 4a^2 times
a.

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00:10:05,000 --> 00:10:11,000
So, I get b2 over 4a y^2.
So, and I want,

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00:10:11,000 --> 00:10:17,000
in fact, c times y^2.
So, the number of y^2 that I

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00:10:17,000 --> 00:10:22,000
should add is c minus b^2 over
4a.

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00:10:22,000 --> 00:10:27,000
OK, let's see that again.
If I expand this thing,

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00:10:27,000 --> 00:10:33,000
I will get ax^2 plus a times b
over 2a times 2xy.

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00:10:33,000 --> 00:10:39,000
That's going to be my bxy.
But, I also get b^2 over 4a^2

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00:10:39,000 --> 00:10:44,000
y^2 times a.
That's b^2 over 4ay^2.

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00:10:44,000 --> 00:10:47,000
And, that cancels out with this
guy here.

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00:10:47,000 --> 00:10:52,000
And then, I will be left with
cy^2.

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00:10:52,000 --> 00:10:58,000
OK, do you see it kind of?
OK, if not, well,

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00:10:58,000 --> 00:11:04,000
try expanding this square
again.

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00:11:04,000 --> 00:11:06,000
OK, maybe I'll do it just to
convince you.

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00:11:06,000 --> 00:11:11,000
But, so if I expand this,
I will get A times,

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00:11:11,000 --> 00:11:16,000
let me put that in a different
color because you shouldn't

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write that down.
It's just to convince you again.

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00:11:19,000 --> 00:11:25,000
So, if you don't see it yet,
let's expend this thing.

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00:11:25,000 --> 00:11:35,000
We'll get a times x^2 plus a
times 2xb over 2ay.

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00:11:35,000 --> 00:11:42,000
Well, the two A's cancel out.
We get bxy plus a times the

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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

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00:11:53,000 --> 00:11:59,000
b^2 over 4ay^2.
Here, the a and the a

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00:11:59,000 --> 00:12:06,000
simplifies, and now these two
terms simplify and give me just

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00:12:06,000 --> 00:12:09,000
cy^2 in the end.
OK, and that's kind of

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00:12:09,000 --> 00:12:12,000
unreadable after I've canceled
everything,

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00:12:12,000 --> 00:12:19,000
but if you follow it,
you see that basically I've

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00:12:19,000 --> 00:12:24,000
just rewritten my initial
function.

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00:12:24,000 --> 00:12:29,000
OK, is that kind of OK?
I mean, otherwise there's just

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00:12:29,000 --> 00:12:32,000
no substitute.
You'll have to do it yourself,

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I'm afraid.
OK, so, let me continue to play

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with this.
So, I'm just going to put this

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in a slightly different form
just to clear the denominators.

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00:12:48,000 --> 00:12:56,000
OK, so, I will instead write
this as one over 4a times the

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00:12:56,000 --> 00:13:03,000
big thing.
So, I'm going to just put 4a^2

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00:13:03,000 --> 00:13:10,000
times x plus b over 2ay squared.
OK, so far I have the same

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00:13:10,000 --> 00:13:13,000
thing as here.
I just introduced the 4a that

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00:13:13,000 --> 00:13:19,000
cancels out, plus for the other
one, I'm just clearing the

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00:13:19,000 --> 00:13:28,000
denominator.
I end up with (4ac-b^2)y^2.

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00:13:28,000 --> 00:13:32,000
OK, so that's a lot of terms.
But, what does it look like?

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00:13:32,000 --> 00:13:35,000
Well, it looks like,
so we have some constant

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00:13:35,000 --> 00:13:38,000
factors, and here we have a
square, and here we have a

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00:13:38,000 --> 00:13:39,000
square.
So, basically,

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00:13:39,000 --> 00:13:44,000
we've written this as a sum of
two squares, well,

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00:13:44,000 --> 00:13:47,000
a sum or a difference of two
squares.

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00:13:47,000 --> 00:13:51,000
And, maybe that's what we need
to figure out to know what kind

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00:13:51,000 --> 00:13:55,000
of point it is because,
see, if you take a sum of two

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00:13:55,000 --> 00:13:57,000
squares,
that you will know that each

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00:13:57,000 --> 00:14:01,000
square takes nonnegative values.
And you will have,

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00:14:01,000 --> 00:14:04,000
the function will always take
nonnegative values.

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00:14:04,000 --> 00:14:07,000
So, the origin will be a
minimum.

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00:14:07,000 --> 00:14:10,000
Well, if you have a difference
of two squares that typically

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00:14:10,000 --> 00:14:13,000
you'll have a saddle point
because depending on whether one

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00:14:13,000 --> 00:14:18,000
or the other is larger,
you will have a positive or a

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00:14:18,000 --> 00:14:24,000
negative quantity.
OK, so I claim there's various

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00:14:24,000 --> 00:14:32,000
cases to look at.
So, let's see.

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00:14:32,000 --> 00:14:34,000
So, in fact,
I claim there will be three

192
00:14:34,000 --> 00:14:37,000
cases.
And, that's good news for us

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00:14:37,000 --> 00:14:40,000
because after all,
we want to distinguish between

194
00:14:40,000 --> 00:14:45,000
three possibilities.
So, let's first do away with

195
00:14:45,000 --> 00:14:52,000
the most complicated one.
What if 4ac minus b^2 is

196
00:14:52,000 --> 00:14:56,000
negative?
Well, if it's negative,

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00:14:56,000 --> 00:15:00,000
then it means what I have
between the brackets is,

198
00:15:00,000 --> 00:15:06,000
so the first guy is obviously a
positive quantity,

199
00:15:06,000 --> 00:15:10,000
while the second one will be
something negative times y2.

200
00:15:10,000 --> 00:15:13,000
So, it will be a negative
quantity.

201
00:15:13,000 --> 00:15:23,000
OK, so one term is positive.
The other is negative.

202
00:15:23,000 --> 00:15:31,000
That tells us we actually have
a saddle point.

203
00:15:31,000 --> 00:15:35,000
We have, in fact,
written our function as a

204
00:15:35,000 --> 00:15:40,000
difference of two squares.
OK, is that convincing?

205
00:15:40,000 --> 00:15:42,000
So, if you want,
what I could do is actually I

206
00:15:42,000 --> 00:15:47,000
could change my coordinates,
have new coordinates called u

207
00:15:47,000 --> 00:15:50,000
equals x b over 2ay,
and v, actually,

208
00:15:50,000 --> 00:15:55,000
well, I could keep y,
and that it would look like the

209
00:15:55,000 --> 00:16:02,000
difference of squares directly.
OK, so that's the first case.

210
00:16:02,000 --> 00:16:12,000
The second case is where
4ac-b^2 = 0.

211
00:16:12,000 --> 00:16:18,000
Well, what happens if that's
zero?

212
00:16:18,000 --> 00:16:21,000
Then it means that this term
over there goes away.

213
00:16:21,000 --> 00:16:25,000
So, what we have is just one
square.

214
00:16:25,000 --> 00:16:29,000
OK, so what that means is
actually that our function

215
00:16:29,000 --> 00:16:32,000
depends only on one direction of
things.

216
00:16:32,000 --> 00:16:36,000
In the other direction,
it's going to actually be

217
00:16:36,000 --> 00:16:38,000
degenerate.
So, for example,

218
00:16:38,000 --> 00:16:40,000
forget all the clutter in
there.

219
00:16:40,000 --> 00:16:45,000
Say I give you just the
function of two variables,

220
00:16:45,000 --> 00:16:49,000
w equals just x^2.
So, that means it doesn't

221
00:16:49,000 --> 00:16:53,000
depend on y at all.
And, if I try to plot the

222
00:16:53,000 --> 00:16:58,000
graph, it will look like,
well, x is here.

223
00:16:58,000 --> 00:17:04,000
So, it will depend on x in that
way, but it doesn't depend on y

224
00:17:04,000 --> 00:17:10,000
at all.
So, what the graph looks like

225
00:17:10,000 --> 00:17:18,000
is something like that.
OK, basically it's a valley

226
00:17:18,000 --> 00:17:22,000
whose bottom is completely flat.
So, that means,

227
00:17:22,000 --> 00:17:24,000
actually, we have a degenerate
critical point.

228
00:17:24,000 --> 00:17:28,000
It's called degenerate because
there is a direction in which

229
00:17:28,000 --> 00:17:30,000
nothing happens.
And, in fact,

230
00:17:30,000 --> 00:17:38,000
you have critical points
everywhere along the y axis.

231
00:17:38,000 --> 00:17:42,000
Now, whether the square that we
have is x or something else,

232
00:17:42,000 --> 00:17:46,000
namely, x plus b over 2a y,
it doesn't matter.

233
00:17:46,000 --> 00:17:48,000
I mean, it will still get this
degenerate behavior.

234
00:17:48,000 --> 00:17:56,000
But there's a direction in
which nothing happens because we

235
00:17:56,000 --> 00:18:02,000
just have the square of one
quantity.

236
00:18:02,000 --> 00:18:06,000
I'm sure that 300 students
means 300 different ring tones,

237
00:18:06,000 --> 00:18:09,000
but I'm not eager to hear all
of them, thanks.

238
00:18:09,000 --> 00:18:18,000
[LAUGHTER]
OK, so, this is what's called a

239
00:18:18,000 --> 00:18:28,000
degenerate critical point,
and [LAUGHTER].

240
00:18:28,000 --> 00:18:33,000
OK, so basically we'll leave it
here.

241
00:18:33,000 --> 00:18:38,000
We won't actually try to figure
out further what happens,

242
00:18:38,000 --> 00:18:42,000
and the reason for that is that
when you have an actual

243
00:18:42,000 --> 00:18:44,000
function,
a general function,

244
00:18:44,000 --> 00:18:46,000
not just one that's quadratic
like this,

245
00:18:46,000 --> 00:18:50,000
then there will actually be
other terms maybe involving

246
00:18:50,000 --> 00:18:54,000
higher powers,
maybe x^3 or y^3 or things like

247
00:18:54,000 --> 00:18:56,000
that.
And then, they will mess up

248
00:18:56,000 --> 00:19:00,000
what happens in this valley.
And, it's a situation where we

249
00:19:00,000 --> 00:19:03,000
won't be able,
actually, to tell automatically

250
00:19:03,000 --> 00:19:06,000
just by looking at second
derivatives what happens.

251
00:19:06,000 --> 00:19:09,000
See, for example,
in a function of one variable,

252
00:19:09,000 --> 00:19:12,000
if you have just a function of
one variable,

253
00:19:12,000 --> 00:19:14,000
say, f of x equals x to the
five,

254
00:19:14,000 --> 00:19:18,000
well, if you try to decide what
type of point the origin is,

255
00:19:18,000 --> 00:19:20,000
you're going to take the second
derivative.

256
00:19:20,000 --> 00:19:23,000
It will be zero,
and then you can conclude.

257
00:19:23,000 --> 00:19:26,000
Those things depend on higher
order derivatives.

258
00:19:26,000 --> 00:19:29,000
So, we just won't like that
case.

259
00:19:29,000 --> 00:19:34,000
We just won't try to figure out
what's going on here.

260
00:19:34,000 --> 00:19:40,000
Now, the last situation is if
4ac-b^2 is positive.

261
00:19:40,000 --> 00:19:45,000
So, then, that means that
actually we've written things.

262
00:19:45,000 --> 00:19:52,000
The big bracket up there is a
sum of two squares.

263
00:19:52,000 --> 00:20:00,000
So, that means that we've
written w as one over 4a times

264
00:20:00,000 --> 00:20:08,000
plus something squared plus
something else squared,

265
00:20:08,000 --> 00:20:12,000
OK?
So, these guys have the same

266
00:20:12,000 --> 00:20:18,000
sign, and that means that this
term here will always be greater

267
00:20:18,000 --> 00:20:22,000
than or equal to zero.
And that means that we should

268
00:20:22,000 --> 00:20:24,000
either have a maximum or
minimum.

269
00:20:24,000 --> 00:20:29,000
How we find out which one it is?
Well, we look at the sign of a,

270
00:20:29,000 --> 00:20:30,000
exactly.
OK?

271
00:20:30,000 --> 00:20:35,000
So, there's two sub-cases.
One is if a is positive,

272
00:20:35,000 --> 00:20:40,000
then, this quantity overall
will always be nonnegative.

273
00:20:40,000 --> 00:20:54,000
And that means we have a
minimum, OK?

274
00:20:54,000 --> 00:20:58,000
And, if a is negative on the
other hand,

275
00:20:58,000 --> 00:21:01,000
so that means that we multiply
this positive quantity by a

276
00:21:01,000 --> 00:21:04,000
negative number,
we get something that's always

277
00:21:04,000 --> 00:21:10,000
negative.
So, zero is actually the

278
00:21:10,000 --> 00:21:18,000
maximum.
OK, is that clear for everyone?

279
00:21:18,000 --> 00:21:29,000
Yes?
Sorry, yeah,

280
00:21:29,000 --> 00:21:34,000
so I said in the example w
equals x^2, it doesn't depend on

281
00:21:34,000 --> 00:21:37,000
y.
So, the more general situation

282
00:21:37,000 --> 00:21:44,000
is w equals some constant.
Well, I guess it's a times (x b

283
00:21:44,000 --> 00:21:48,000
over 2a times y)^2.
So, it does depend on x and y,

284
00:21:48,000 --> 00:21:51,000
but it only depends on this
combination.

285
00:21:51,000 --> 00:21:54,000
OK, so if I choose to move in
some other perpendicular

286
00:21:54,000 --> 00:21:58,000
direction,
in the direction where this

287
00:21:58,000 --> 00:22:02,000
remains constant,
so maybe if I set x equals

288
00:22:02,000 --> 00:22:06,000
minus b over 2a y,
then this remains zero all the

289
00:22:06,000 --> 00:22:08,000
time.
So, there's a degenerate

290
00:22:08,000 --> 00:22:11,000
direction in which I stay at the
minimum or maximum,

291
00:22:11,000 --> 00:22:15,000
or whatever it is that I have.
OK, so that's why it's called

292
00:22:15,000 --> 00:22:17,000
degenerate.
There is a direction in which

293
00:22:17,000 --> 00:22:29,000
nothing happens.
OK, yes?

294
00:22:29,000 --> 00:22:31,000
Yes, yeah, so that's a very
good question.

295
00:22:31,000 --> 00:22:33,000
So, there's going to be the
second derivative test.

296
00:22:33,000 --> 00:22:36,000
Why do not have derivatives yet?
Well, that's because I've been

297
00:22:36,000 --> 00:22:39,000
looking at this special example
where we have a function like

298
00:22:39,000 --> 00:22:41,000
this.
And, so I don't actually need

299
00:22:41,000 --> 00:22:43,000
to take derivatives yet.
But, secretly,

300
00:22:43,000 --> 00:22:46,000
that's because a,
b, and c will be the second

301
00:22:46,000 --> 00:22:49,000
derivatives of the function,
actually, 2a,

302
00:22:49,000 --> 00:22:52,000
b, and 2c.
So now, we are going to go to

303
00:22:52,000 --> 00:22:54,000
general function.
And there, instead of having

304
00:22:54,000 --> 00:22:57,000
these coefficients a,
b, and c given to us,

305
00:22:57,000 --> 00:23:00,000
we'll have to compute them as
second derivatives.

306
00:23:00,000 --> 00:23:03,000
OK, so here,
I'm basically setting the stage

307
00:23:03,000 --> 00:23:07,000
for what will be the actual
criterion we'll use using second

308
00:23:07,000 --> 00:23:13,000
derivatives.
Yes?

309
00:23:13,000 --> 00:23:16,000
So, yeah, so what you have a
degenerate critical point,

310
00:23:16,000 --> 00:23:20,000
it could be a degenerate
minimum, or a degenerate maximum

311
00:23:20,000 --> 00:23:23,000
depending on the sign of a.
But, in general,

312
00:23:23,000 --> 00:23:26,000
once you start having
functions, you don't really know

313
00:23:26,000 --> 00:23:30,000
what will happen anymore.
It could also be a degenerate

314
00:23:30,000 --> 00:23:36,000
saddle, and so on.
So, we won't really be able to

315
00:23:36,000 --> 00:23:40,000
tell.
Yes?

316
00:23:40,000 --> 00:23:43,000
It is possible to have a
degenerate saddle point.

317
00:23:43,000 --> 00:23:46,000
For example,
if I gave you x^3 y^3,

318
00:23:46,000 --> 00:23:49,000
you can convince yourself that
if you take x and y to be

319
00:23:49,000 --> 00:23:53,000
negative, it will be negative.
If x and y are positive,

320
00:23:53,000 --> 00:23:55,000
it's positive.
And, it has a very degenerate

321
00:23:55,000 --> 00:23:59,000
critical point at the origin.
So, that's a degenerate saddle

322
00:23:59,000 --> 00:24:01,000
point.
We don't see it here because

323
00:24:01,000 --> 00:24:04,000
that doesn't happen if you have
only quadratic terms like that.

324
00:24:04,000 --> 00:24:12,000
You need to have higher-order
terms to see it happen.

325
00:24:12,000 --> 00:24:23,000
OK.
OK, so let's continue.

326
00:24:23,000 --> 00:24:27,000
Before we continue,
but see, I wanted to point out

327
00:24:27,000 --> 00:24:30,000
one small thing.
So, here, we have the magic

328
00:24:30,000 --> 00:24:34,000
quantity, 4ac minus b^2.
You've probably seen that

329
00:24:34,000 --> 00:24:37,000
before in your life.
Yet, it looks like the

330
00:24:37,000 --> 00:24:40,000
quadratic formula,
except that one involves

331
00:24:40,000 --> 00:24:43,000
b^2-4ac.
But that's really the same

332
00:24:43,000 --> 00:24:47,000
thing.
OK, so let's see,

333
00:24:47,000 --> 00:24:57,000
where does the quadratic
formula come in here?

334
00:24:57,000 --> 00:25:00,000
Well, let me write things
differently.

335
00:25:00,000 --> 00:25:03,000
OK, so we've manipulated
things, and got into a

336
00:25:03,000 --> 00:25:08,000
conclusion.
But, let me just do a different

337
00:25:08,000 --> 00:25:14,000
manipulation,
and write this now instead as

338
00:25:14,000 --> 00:25:23,000
y^2 times a times x over y
squared plus b(x over y) plus c.

339
00:25:23,000 --> 00:25:28,000
OK, see, that's the same thing
that I had before.

340
00:25:28,000 --> 00:25:35,000
Well, so now this quantity here
is always nonnegative.

341
00:25:35,000 --> 00:25:39,000
What about this one?
Well, of course,

342
00:25:39,000 --> 00:25:43,000
this one depends on x over y.
It means it depends on which

343
00:25:43,000 --> 00:25:45,000
direction you're going to move
away from the origin,

344
00:25:45,000 --> 00:25:48,000
which ratio between x and y you
will consider.

345
00:25:48,000 --> 00:25:51,000
But, I claim there's two
situations.

346
00:25:51,000 --> 00:25:57,000
One is, so, let's try to
reformulate things.

347
00:25:57,000 --> 00:26:04,000
So, if a discriminate here is
positive, then it means that

348
00:26:04,000 --> 00:26:10,000
these have roots and these have
solutions.

349
00:26:10,000 --> 00:26:19,000
And, that means that this
quantity can be both positive

350
00:26:19,000 --> 00:26:24,000
and negative.
This quantity takes positive

351
00:26:24,000 --> 00:26:31,000
and negative values.
One way to convince yourself is

352
00:26:31,000 --> 00:26:37,000
just to, you know,
plot at^2 bt c.

353
00:26:37,000 --> 00:26:43,000
You know that there's two roots.
So, it might look like this,

354
00:26:43,000 --> 00:26:48,000
or might look like that
depending on the sign of a.

355
00:26:48,000 --> 00:26:52,000
But, in either case,
it will take values of both

356
00:26:52,000 --> 00:26:54,000
signs.
So, that means that your

357
00:26:54,000 --> 00:26:56,000
function will take values of
both signs.

358
00:27:04,000 --> 00:27:13,000
The value takes both positive
and negative values.

359
00:27:13,000 --> 00:27:21,000
And, so that means we have a
saddle point,

360
00:27:21,000 --> 00:27:28,000
while the other situation,
when b^2-4ac is negative -- --

361
00:27:28,000 --> 00:27:36,000
means that this equation is
quadratic never takes the value,

362
00:27:36,000 --> 00:27:39,000
zero.
So, it's always positive or

363
00:27:39,000 --> 00:27:42,000
it's always negative,
depending on the sign of a.

364
00:27:42,000 --> 00:27:48,000
So, the other case is if
b^2-4ac is negative,

365
00:27:48,000 --> 00:27:53,000
then the quadratic doesn't have
a solution.

366
00:27:53,000 --> 00:27:58,000
And it could look like this or
like that depending on whether a

367
00:27:58,000 --> 00:28:03,000
is positive or a is negative.
So, in particular,

368
00:28:03,000 --> 00:28:12,000
that means that ax over y2 plus
bx over y plus c is always

369
00:28:12,000 --> 00:28:21,000
positive or always negative
depending on the sign of a.

370
00:28:21,000 --> 00:28:23,000
And then, that tells us that
our function,

371
00:28:23,000 --> 00:28:25,000
w, will be always positive or
always negative.

372
00:28:25,000 --> 00:28:28,000
And then we'll get a minimum or
maximum.

373
00:28:40,000 --> 00:28:44,000
OK, we'll have a min or a max
depending on which situation we

374
00:28:44,000 --> 00:28:47,000
are in.
OK, so that's another way to

375
00:28:47,000 --> 00:28:51,000
derive the same answer.
And now, you see here why the

376
00:28:51,000 --> 00:28:55,000
discriminate plays a role.
That's because it exactly tells

377
00:28:55,000 --> 00:28:59,000
you whether this quadratic
quantity has always the same

378
00:28:59,000 --> 00:29:04,000
sign,
or whether it can actually

379
00:29:04,000 --> 00:29:12,000
cross the value,
zero, when you have the root of

380
00:29:12,000 --> 00:29:16,000
a quadratic.
OK, so hopefully at this stage

381
00:29:16,000 --> 00:29:20,000
you are happy with one of the
two explanations,

382
00:29:20,000 --> 00:29:23,000
at least.
And now, you are willing to

383
00:29:23,000 --> 00:29:26,000
believe, I hope,
that we have basically a way of

384
00:29:26,000 --> 00:29:30,000
deciding what type of critical
point we have in the special

385
00:29:30,000 --> 00:29:32,000
case of a quadratic function.

386
00:29:58,000 --> 00:30:05,000
OK, so, now what do we do with
the general function?

387
00:30:05,000 --> 00:30:19,000
Well, so in general,
we want to look at second

388
00:30:19,000 --> 00:30:24,000
derivatives.
OK, so now we are getting to

389
00:30:24,000 --> 00:30:27,000
the real stuff.
So, how many second derivatives

390
00:30:27,000 --> 00:30:29,000
do we have?
That's maybe the first thing we

391
00:30:29,000 --> 00:30:32,000
should figure out.
Well, we can take the

392
00:30:32,000 --> 00:30:39,000
derivative first with respect to
x, and then again with respect

393
00:30:39,000 --> 00:30:44,000
to x.
OK, that gives us something we

394
00:30:44,000 --> 00:30:54,000
denote by partial square f over
partial x squared or fxx.

395
00:30:54,000 --> 00:31:00,000
Then, there's another one which
is fxy, which means you take the

396
00:31:00,000 --> 00:31:05,000
derivative with respect to x,
and then with respect to y.

397
00:31:05,000 --> 00:31:09,000
Another thing you can do,
is do first derivative respect

398
00:31:09,000 --> 00:31:12,000
to y, and then with respect to
x.

399
00:31:12,000 --> 00:31:17,000
That would be fyx.
Well, good news.

400
00:31:17,000 --> 00:31:22,000
These are actually always equal
to each other.

401
00:31:22,000 --> 00:31:26,000
OK, so it's the fact that we
will admit, it's actually not

402
00:31:26,000 --> 00:31:30,000
very hard to check.
So these are always the same.

403
00:31:30,000 --> 00:31:33,000
We don't need to worry about
which one we do.

404
00:31:33,000 --> 00:31:36,000
That's one computation that we
won't need to do.

405
00:31:36,000 --> 00:31:43,000
We can save a bit of effort.
And then, we have the last one,

406
00:31:43,000 --> 00:31:51,000
namely, the second partial with
respect to y and y fyy.

407
00:31:51,000 --> 00:32:00,000
OK, so we have three of them.
So, what does the second

408
00:32:00,000 --> 00:32:02,000
derivative test say?

409
00:32:16,000 --> 00:32:22,000
It says, say that you have a
critical point (x0,

410
00:32:22,000 --> 00:32:27,000
y0) of a function of two
variables, f,

411
00:32:27,000 --> 00:32:34,000
and then let's compute the
partial derivatives.

412
00:32:34,000 --> 00:32:41,000
So, let's call capital A the
second derivative with respect

413
00:32:41,000 --> 00:32:45,000
to x.
Let's call capital B the second

414
00:32:45,000 --> 00:32:49,000
derivative with respect to x and
y.

415
00:32:49,000 --> 00:32:55,000
And C equals fyy at this point,
OK?

416
00:32:55,000 --> 00:32:59,000
So, these are just numbers
because we first compute the

417
00:32:59,000 --> 00:33:02,000
second derivative,
and then we plug in the values

418
00:33:02,000 --> 00:33:04,000
of x and y at the critical
point.

419
00:33:04,000 --> 00:33:14,000
So, these will just be numbers.
And now, what we do is we look

420
00:33:14,000 --> 00:33:21,000
at the quantity AC-B^2.
I am not forgetting the four.

421
00:33:21,000 --> 00:33:26,000
You will see why there isn't
one.

422
00:33:26,000 --> 00:33:31,000
So, if AC-B^2 is positive,
then there's two sub-cases.

423
00:33:31,000 --> 00:33:39,000
If A is positive,
then it's local minimum.

424
00:33:50,000 --> 00:33:56,000
The second case,
so, still, if AC-B^2 is

425
00:33:56,000 --> 00:34:04,000
positive, but A is negative,
then it's going to be a local

426
00:34:04,000 --> 00:34:11,000
maximum.
And, if AC-B^2 is negative,

427
00:34:11,000 --> 00:34:17,000
then it's a saddle point,
and finally,

428
00:34:17,000 --> 00:34:24,000
if AC-B^2 is zero,
then we actually cannot

429
00:34:24,000 --> 00:34:28,000
compute.
We don't know whether it's

430
00:34:28,000 --> 00:34:33,000
going to be a minimum,
a maximum, or a saddle.

431
00:34:33,000 --> 00:34:37,000
We know it's degenerate in some
way, but we don't know what type

432
00:34:37,000 --> 00:34:40,000
of point it is.
OK, so that's actually what you

433
00:34:40,000 --> 00:34:43,000
need to remember.
If you are formula oriented,

434
00:34:43,000 --> 00:34:45,000
that's all you need to remember
about today.

435
00:34:45,000 --> 00:34:53,000
But, let's try to understand
why, how this comes out of what

436
00:34:53,000 --> 00:34:59,000
we had there.
OK, so, I think maybe I

437
00:34:59,000 --> 00:35:05,000
actually want to keep,
so maybe I want to keep this

438
00:35:05,000 --> 00:35:06,000
middle board because it actually
has,

439
00:35:06,000 --> 00:35:09,000
you know, the recipe that we
found before the quadratic

440
00:35:09,000 --> 00:35:12,000
function.
So, let me move directly over

441
00:35:12,000 --> 00:35:16,000
there and try to relate our old
recipe with the new.

442
00:35:43,000 --> 00:35:50,000
OK, you are easily amused.
OK, so first,

443
00:35:50,000 --> 00:35:57,000
let's check that these two
things say the same thing in the

444
00:35:57,000 --> 00:36:01,000
special case that we are looking
at.

445
00:36:01,000 --> 00:36:12,000
OK, so let's verify in the
special case where the function

446
00:36:12,000 --> 00:36:22,000
was ax^2 bxy cy^2.
So -- Well, what is the second

447
00:36:22,000 --> 00:36:28,000
derivative with respect to x and
x?

448
00:36:28,000 --> 00:36:31,000
If I take the second derivative
with respect to x and x,

449
00:36:31,000 --> 00:36:34,000
so first I want to take maybe
the derivative with respect to

450
00:36:34,000 --> 00:36:37,000
x.
But first, let's take the first

451
00:36:37,000 --> 00:36:46,000
partial, Wx.
That will be 2ax by, right?

452
00:36:46,000 --> 00:36:50,000
So, Wxx will be,
well, let's take a partial with

453
00:36:50,000 --> 00:36:54,000
respect to x again.
That's 2a.

454
00:36:54,000 --> 00:37:02,000
Wxy, I take the partial respect
to y, and we'll get b.

455
00:37:02,000 --> 00:37:06,000
OK, now we need,
also, the partial with respect

456
00:37:06,000 --> 00:37:13,000
to y.
So, Wy is bx 2cy.

457
00:37:13,000 --> 00:37:17,000
In case you don't believe what
I told you about the mixed

458
00:37:17,000 --> 00:37:21,000
partials, Wyx,
well, you can check.

459
00:37:21,000 --> 00:37:24,000
And it's, again, b.
So, they are,

460
00:37:24,000 --> 00:37:30,000
indeed, the same thing.
And, Wyy will be 2c.

461
00:37:30,000 --> 00:37:39,000
So, if we now look at these
quantities, that tells us,

462
00:37:39,000 --> 00:37:46,000
well, big A is two little a,
big B is little b,

463
00:37:46,000 --> 00:37:55,000
big C is two little c.
So, AC-B^2 is what we used to

464
00:37:55,000 --> 00:38:04,000
call four little ac minus b2.
OK, ooh.

465
00:38:04,000 --> 00:38:07,000
[LAUGHTER]
So, now you can compare the

466
00:38:07,000 --> 00:38:10,000
cases.
They are not listed in the same

467
00:38:10,000 --> 00:38:14,000
order just to make it harder.
So, we said first,

468
00:38:14,000 --> 00:38:20,000
so the saddle case is when
AC-B^2 in big letters is

469
00:38:20,000 --> 00:38:26,000
negative, that's the same as
4ac-b2 in lower case is

470
00:38:26,000 --> 00:38:30,000
negative.
The case where capital AC-B2 is

471
00:38:30,000 --> 00:38:35,000
positive, local min and local
max corresponds to this one.

472
00:38:35,000 --> 00:38:40,000
And, the case where we can't
conclude was what used to be the

473
00:38:40,000 --> 00:38:44,000
degenerate one.
OK, so at least we don't seem

474
00:38:44,000 --> 00:38:48,000
to have messed up when copying
the formula.

475
00:38:48,000 --> 00:38:56,000
Now, why does that work more
generally than that?

476
00:38:56,000 --> 00:39:03,000
Well, the answer that is,
again, Taylor approximation.

477
00:39:03,000 --> 00:39:16,000
Aww.
OK, so let me just do here

478
00:39:16,000 --> 00:39:22,000
quadratic approximation.
So, quadratic approximation

479
00:39:22,000 --> 00:39:25,000
tells me the following thing.
It tells me,

480
00:39:25,000 --> 00:39:30,000
if I have a function,
f of xy, and I want to

481
00:39:30,000 --> 00:39:37,000
understand the change in f when
I change x and y a little bit.

482
00:39:37,000 --> 00:39:40,000
Well, there's the first-order
terms.

483
00:39:40,000 --> 00:39:43,000
There is the linear terms that
by now you should know and be

484
00:39:43,000 --> 00:39:51,000
comfortable with.
That's fx times the change in x.

485
00:39:51,000 --> 00:39:56,000
And then, there's fy times the
change in y.

486
00:39:56,000 --> 00:40:00,000
OK, that's the starting point.
But now, of course,

487
00:40:00,000 --> 00:40:03,000
if x and y, sorry,
if we are at the critical

488
00:40:03,000 --> 00:40:09,000
point, then that's going to be
zero at the critical point.

489
00:40:09,000 --> 00:40:16,000
So, that term actually goes
away, and that's also zero at

490
00:40:16,000 --> 00:40:22,000
the critical point.
So, that term also goes away.

491
00:40:22,000 --> 00:40:24,000
OK, so linear approximation is
really no good.

492
00:40:24,000 --> 00:40:27,000
We need more terms.
So, what are the next terms?

493
00:40:27,000 --> 00:40:35,000
Well, the next terms are
quadratic terms,

494
00:40:35,000 --> 00:40:38,000
and so I mean,
if you remember the Taylor

495
00:40:38,000 --> 00:40:42,000
formula for a function of a
single variable,

496
00:40:42,000 --> 00:40:46,000
there was the derivative times
x minus x0 plus one half of a

497
00:40:46,000 --> 00:40:51,000
second derivative times x-x0^2.
And see, this side here is

498
00:40:51,000 --> 00:40:55,000
really Taylor approximation in
one variable looking only at x.

499
00:40:55,000 --> 00:40:57,000
But of course,
we also have terms involving y,

500
00:40:57,000 --> 00:41:00,000
and terms involving
simultaneously x and y.

501
00:41:00,000 --> 00:41:10,000
And, these terms are fxy times
change in x times change in y

502
00:41:10,000 --> 00:41:17,000
plus one half of fyy(y-y0)^2.
There's no one half in the

503
00:41:17,000 --> 00:41:20,000
middle because,
in fact, you would have two

504
00:41:20,000 --> 00:41:24,000
terms, one for xy,
one for yx, but they are the

505
00:41:24,000 --> 00:41:26,000
same.
And then, if you want to

506
00:41:26,000 --> 00:41:29,000
continue, there is actually
cubic terms involving the third

507
00:41:29,000 --> 00:41:32,000
derivatives, and so on,
but we are not actually looking

508
00:41:32,000 --> 00:41:34,000
at them.
And so, now,

509
00:41:34,000 --> 00:41:39,000
when we do this approximation,
well, the type of critical

510
00:41:39,000 --> 00:41:45,000
point remains the same when we
replace the function by this

511
00:41:45,000 --> 00:41:48,000
approximation.
And so, we can apply the

512
00:41:48,000 --> 00:41:53,000
argument that we used to deduce
things in the quadratic case.

513
00:41:53,000 --> 00:41:55,000
In fact, it still works in the
general case using this

514
00:41:55,000 --> 00:41:57,000
approximation formula.

515
00:42:12,000 --> 00:42:26,000
So -- The general case reduces
to the quadratic case.

516
00:42:26,000 --> 00:42:31,000
And now, you see actually why,
well, here you see,

517
00:42:31,000 --> 00:42:36,000
again, how this coefficient
which we used to call little a

518
00:42:36,000 --> 00:42:41,000
is also one half of capital A.
And same here:

519
00:42:41,000 --> 00:42:47,000
this coefficient is what we
call capital B or little b,

520
00:42:47,000 --> 00:42:52,000
and this coefficient here is
what we called little c or one

521
00:42:52,000 --> 00:42:57,000
half of capital C.
And then, when you replace

522
00:42:57,000 --> 00:43:02,000
these into the various cases
that we had here,

523
00:43:02,000 --> 00:43:06,000
you end up with the second
derivative test.

524
00:43:06,000 --> 00:43:08,000
So, what about the degenerate
case?

525
00:43:08,000 --> 00:43:11,000
Why can't we just say,
well, it's going to be a

526
00:43:11,000 --> 00:43:16,000
degenerate critical point?
So, the reason is that this

527
00:43:16,000 --> 00:43:20,000
approximation formula is
reasonable only if the higher

528
00:43:20,000 --> 00:43:24,000
order terms are negligible.
OK, so in fact,

529
00:43:24,000 --> 00:43:27,000
secretly, there's more terms.
This is only an approximation.

530
00:43:27,000 --> 00:43:30,000
There would be terms involving
third derivatives,

531
00:43:30,000 --> 00:43:34,000
and maybe even beyond that.
And, so it is not to generate

532
00:43:34,000 --> 00:43:37,000
case,
they don't actually matter

533
00:43:37,000 --> 00:43:39,000
because the shape of the
function,

534
00:43:39,000 --> 00:43:42,000
the shape of the graph,
is actually determined by the

535
00:43:42,000 --> 00:43:45,000
quadratic terms.
But, in the degenerate case,

536
00:43:45,000 --> 00:43:49,000
see, if I start with this and I
add something even very,

537
00:43:49,000 --> 00:43:53,000
very small along the y axis,
then that can be enough to bend

538
00:43:53,000 --> 00:43:56,000
this very slightly up or
slightly down,

539
00:43:56,000 --> 00:44:00,000
and turn my degenerate point in
to either a minimum or a saddle

540
00:44:00,000 --> 00:44:03,000
point.
And, I won't be able to tell

541
00:44:03,000 --> 00:44:06,000
until I go further in the list
of derivatives.

542
00:44:06,000 --> 00:44:14,000
So, in the degenerate case,
what actually happens depends

543
00:44:14,000 --> 00:44:20,000
on the higher order derivatives.

544
00:44:38,000 --> 00:44:42,000
So, we will need to analyze
things more carefully.

545
00:44:42,000 --> 00:44:45,000
Well, we're not going to bother
with that in this class.

546
00:44:45,000 --> 00:44:52,000
So, we'll just say,
well, we cannot compute,

547
00:44:52,000 --> 00:44:54,000
OK?
I mean, you have to realize

548
00:44:54,000 --> 00:44:57,000
that in real life,
you have to be extremely

549
00:44:57,000 --> 00:45:02,000
unlucky for this quantity to end
up being exactly 0.

550
00:45:02,000 --> 00:45:03,000
[LAUGHTER]
Well, if that happens,

551
00:45:03,000 --> 00:45:05,000
then what you should do is
maybe try by inspection.

552
00:45:05,000 --> 00:45:08,000
See if there's a good reason
why the function should always

553
00:45:08,000 --> 00:45:11,000
be positive or always be
negative, or something.

554
00:45:11,000 --> 00:45:16,000
Or, you know,
plot it on a computer and see

555
00:45:16,000 --> 00:45:23,000
what happens.
But, otherwise we can't compute.

556
00:45:23,000 --> 00:45:33,000
OK, so let's do an example.
So, probably I should leave

557
00:45:33,000 --> 00:45:39,000
this on so that we still have
the test with us.

558
00:45:39,000 --> 00:45:42,000
And, instead,
OK, so I'll do my example here.

559
00:46:20,000 --> 00:46:30,000
OK, so just an example.
Let's look at f of (x,

560
00:46:30,000 --> 00:46:37,000
y) = x y 1/xy,
where x and y are positive.

561
00:46:37,000 --> 00:46:39,000
So, I'm looking only at the
first quadrant.

562
00:46:39,000 --> 00:46:42,000
OK, I mean, I'm doing this
because I don't want the

563
00:46:42,000 --> 00:46:46,000
denominator to become zero.
So, I'm just looking at the

564
00:46:46,000 --> 00:46:50,000
situation.
So, let's look first for,

565
00:46:50,000 --> 00:46:55,000
so, the question will be,
what are the minimum and

566
00:46:55,000 --> 00:47:03,000
maximum of this function?
So, the first thing we should

567
00:47:03,000 --> 00:47:12,000
do to answer this question is
look for critical points,

568
00:47:12,000 --> 00:47:15,000
OK?
So, for that,

569
00:47:15,000 --> 00:47:19,000
we have to compute the first
derivatives.

570
00:47:19,000 --> 00:47:34,000
OK, so fx is one minus one over
x^2y, OK?

571
00:47:34,000 --> 00:47:39,000
Take the derivative of one over
x, that's negative one over x^2.

572
00:47:39,000 --> 00:47:44,000
And, we'll want to set that
equal to zero.

573
00:47:44,000 --> 00:47:50,000
And fy is one minus one over
xy^2.

574
00:47:50,000 --> 00:47:54,000
And, we want to set that equal
to zero.

575
00:47:54,000 --> 00:47:59,000
So, what are the equations we
have to solve?

576
00:47:59,000 --> 00:48:05,000
Well, I guess x^2y equals one,
I mean, if I move this guy over

577
00:48:05,000 --> 00:48:09,000
here I get one over x^2y equals
one.

578
00:48:09,000 --> 00:48:14,000
That's x^2y equals one,
and xy^2 equals one.

579
00:48:14,000 --> 00:48:18,000
What do you get by comparing
these two?

580
00:48:18,000 --> 00:48:21,000
Well, x and y should both be,
OK, so yeah,

581
00:48:21,000 --> 00:48:24,000
I agree with you that one and
one is a solution.

582
00:48:24,000 --> 00:48:27,000
Why is it the only one?
So, first, if I divide this one

583
00:48:27,000 --> 00:48:29,000
by that one, I get x over y
equals one.

584
00:48:29,000 --> 00:48:34,000
So, it tells me x equals y.
And then, if x equals y,

585
00:48:34,000 --> 00:48:40,000
then if I put that into here,
it will give me y^3 equals one,

586
00:48:40,000 --> 00:48:44,000
which tells me y equals one,
and therefore,

587
00:48:44,000 --> 00:48:50,000
x equals one as well.
OK, so, there's only one

588
00:48:50,000 --> 00:48:54,000
solution.
There's only one critical

589
00:48:54,000 --> 00:48:58,000
point, which is going to be
(1,1).

590
00:48:58,000 --> 00:49:09,000
OK, so, now here's where you do
a bit of work.

591
00:49:09,000 --> 00:49:18,000
What do you think of that
critical point?

592
00:49:18,000 --> 00:49:25,000
OK, I see some valid votes.
I see some, OK,

593
00:49:25,000 --> 00:49:28,000
I see a lot of people answering
four.

594
00:49:28,000 --> 00:49:30,000
[LAUGHTER]
that seems to suggest that

595
00:49:30,000 --> 00:49:34,000
maybe you haven't completed the
second derivative yet.

596
00:49:34,000 --> 00:49:37,000
Yes, I see someone giving the
correct answer.

597
00:49:37,000 --> 00:49:41,000
I see some people not giving
quite the correct answer.

598
00:49:41,000 --> 00:49:43,000
I see more and more correct
answers.

599
00:49:43,000 --> 00:49:49,000
OK, so let's see.
To figure out what type of

600
00:49:49,000 --> 00:49:52,000
point is, we should compute the
second partial derivatives.

601
00:49:52,000 --> 00:50:02,000
So, fxx is, what do we get what
we take the derivative of this

602
00:50:02,000 --> 00:50:11,000
with respect to x?
Two over x^3y, OK?

603
00:50:11,000 --> 00:50:25,000
So, at our point, a will be 2.
Fxy will be one over x^2y^2.

604
00:50:25,000 --> 00:50:37,000
So, B will be one.
And, Fyy is going to be two

605
00:50:37,000 --> 00:50:42,000
over xy^3.
So, C will be two.

606
00:50:42,000 --> 00:50:51,000
And so that tells us,
well, AC-B^2 is four minus one.

607
00:50:51,000 --> 00:51:02,000
Sorry, I should probably use a
different blackboard for that.

608
00:51:02,000 --> 00:51:06,000
AC-B2 is two times two minus
1^2 is three.

609
00:51:06,000 --> 00:51:10,000
It's positive.
That tells us we are either a

610
00:51:10,000 --> 00:51:17,000
local minimum or local maximum.
And, A is positive.

611
00:51:17,000 --> 00:51:21,000
So, it's a local minimum.
And, in fact,

612
00:51:21,000 --> 00:51:23,000
you can check it's the global
minimum.

613
00:51:23,000 --> 00:51:29,000
What about the maximum?
Well, if a maximum is not

614
00:51:29,000 --> 00:51:32,000
actually at a critical point,
it's on the boundary,

615
00:51:32,000 --> 00:51:35,000
or at infinity.
See, so we have actually to

616
00:51:35,000 --> 00:51:39,000
check what happens when x and y
go to zero or to infinity.

617
00:51:39,000 --> 00:51:42,000
Well, if that happens,
if x or y goes to infinity,

618
00:51:42,000 --> 00:51:44,000
then the function goes to
infinity.

619
00:51:44,000 --> 00:51:48,000
Also, if x or y goes to zero,
then one over xy goes to

620
00:51:48,000 --> 00:51:51,000
infinity.
So, the maximum,

621
00:51:51,000 --> 00:51:59,000
well, the function goes to
infinity when x goes to infinity

622
00:51:59,000 --> 00:52:05,000
or y goes to infinity,
or x and y go to zero.

623
00:52:05,000 --> 00:52:07,000
So, it's not at a critical
point.

624
00:52:07,000 --> 00:52:10,000
OK, so, in general,
we have to check both the

625
00:52:10,000 --> 00:52:13,000
critical points and the
boundaries to decide what

626
00:52:13,000 --> 00:52:15,000
happens.
OK, the end.

627
00:52:15,000 --> 00:52:18,000
Have a nice weekend.