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So, today we are going to
continue looking at critical
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00:00:28,000 --> 00:00:31,000
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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00:00:37,000 --> 00:00:41,000
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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00:01:11,000 --> 00:01:15,000
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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00:01:24,000 --> 00:01:28,000
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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00:01:35,000 --> 00:01:37,000
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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00:01:41,000 --> 00:01:45,000
complicated.
It will have several critical
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points.
So, this example here,
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00:01:48,000 --> 00:01:54,000
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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00:02:07,000 --> 00:02:12,000
level curves become circles that
now down and shrink to the
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00:02:12,000 --> 00:02:15,000
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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00:02:51,000 --> 00:02:53,000
Just, of course,
in one case,
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the values increase,
and in another one they
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00:02:56,000 --> 00:03:03,000
decrease.
So, the question -- -- is,
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how do we decide -- -- between
the various possibilities?
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00:03:17,000 --> 00:03:23,000
So, local minimum,
local maximum,
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00:03:23,000 --> 00:03:26,000
or saddle point.
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00:03:38,000 --> 00:03:44,000
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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00:04:05,000 --> 00:04:07,000
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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00:04:14,000 --> 00:04:16,000
what is the maximum of this
function?
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Well, we have two obvious
candidates.
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00:04:19,000 --> 00:04:22,000
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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00:04:24,000 --> 00:04:26,000
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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00:04:40,000 --> 00:04:43,000
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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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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00:04:59,000 --> 00:05:01,000
point.
It could also be,
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00:05:01,000 --> 00:05:05,000
somehow, on the side in some
limiting situation where one
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00:05:05,000 --> 00:05:09,000
variable stops being in the
allowed rang of values or goes
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00:05:09,000 --> 00:05:13,000
to infinity.
So, we have to actually check
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00:05:13,000 --> 00:05:19,000
the boundary and the infinity
behavior of our function to know
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00:05:19,000 --> 00:05:23,000
where, actually,
the minimum and maximum will
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00:05:23,000 --> 00:05:27,000
be.
So, in general,
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00:05:27,000 --> 00:05:37,000
I should point out,
these should occur either at
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00:05:37,000 --> 00:05:48,000
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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00:05:52,000 --> 00:05:55,000
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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00:06:00,000 --> 00:06:04,000
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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00:06:09,000 --> 00:06:16,000
the critical point?
So, we'll use something that's
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00:06:16,000 --> 00:06:21,000
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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00:06:25,000 --> 00:06:29,000
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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00:06:41,000 --> 00:06:44,000
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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00:06:48,000 --> 00:06:52,000
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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00:06:56,000 --> 00:07:01,000
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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00:07:06,000 --> 00:07:10,000
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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00:07:15,000 --> 00:07:20,000
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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00:07:32,000 --> 00:07:36,000
you take the derivative with
respect to x,
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00:07:36,000 --> 00:07:38,000
well, and if you plug x equals
y equals zero,
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00:07:38,000 --> 00:07:42,000
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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00:07:44,000 --> 00:07:47,000
approximation of this,
well, all these guys are much
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00:07:47,000 --> 00:07:50,000
smaller than x and y when x and
y are small.
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00:07:50,000 --> 00:07:55,000
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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00:08:44,000 --> 00:08:47,000
matter what the values of a,
b, and c are.
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00:08:47,000 --> 00:08:50,000
We'll just try to first
complete things to a square.
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00:08:50,000 --> 00:08:55,000
OK, so let's do that.
So, in general,
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00:08:55,000 --> 00:09:01,000
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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00:09:25,000 --> 00:09:28,000
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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00:09:31,000 --> 00:09:39,000
So, what is it the square of?
Well, you'd start with x plus I
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00:09:39,000 --> 00:09:45,000
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.
149
00:11:04,000 --> 00:11:06,000
OK, maybe I'll do it just to
convince you.
150
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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00:11:16,000 --> 00:11:19,000
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,
161
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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00:12:32,000 --> 00:12:38,000
I'm afraid.
OK, so, let me continue to play
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00:12:38,000 --> 00:12:43,000
with this.
So, I'm just going to put this
167
00:12:43,000 --> 00:12:48,000
in a slightly different form
just to clear the denominators.
168
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.
174
00:13:28,000 --> 00:13:32,000
OK, so that's a lot of terms.
But, what does it look like?
175
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,
178
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.
180
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,
184
00:14:01,000 --> 00:14:04,000
the function will always take
nonnegative values.
185
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
188
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.
191
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
193
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,
197
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.