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Last time we saw things about
gradients and directional

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derivatives.
Before that we studied how to

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look for minima and maxima of
functions of several variables.

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00:00:37,000 --> 00:00:41,000
And today we are going to look
again at min/max problems but in

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a different setting,
namely, one for variables that

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are not independent.
And so what we will see is you

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may have heard of Lagrange
multipliers.

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00:00:52,000 --> 00:00:59,000
And this is the one point in
the term when I can shine with

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my French accent and say
Lagrange's name properly.

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00:01:05,000 --> 00:01:08,000
OK.
What are Lagrange multipliers

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00:01:08,000 --> 00:01:13,000
about?
Well, the goal is to minimize

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or maximize a function of
several variables.

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00:01:19,000 --> 00:01:22,000
Let's say, for example,
f of x, y, z,

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but where these variables are
no longer independent.

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00:01:41,000 --> 00:01:43,000
They are not independent.
That means that there is a

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00:01:43,000 --> 00:01:47,000
relation between them.
The relation is maybe some

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00:01:47,000 --> 00:01:52,000
equation of the form g of x,
y, z equals some constant.

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00:01:52,000 --> 00:01:57,000
You take the relation between
x, y, z, you call that g and

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00:01:57,000 --> 00:02:02,000
that gives you the constraint.
And your goal is to minimize f

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00:02:02,000 --> 00:02:05,000
only of those values of x,
y, z that satisfy the

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00:02:05,000 --> 00:02:07,000
constraint.
What is one way to do that?

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00:02:07,000 --> 00:02:10,000
Well, one to do that,
if the constraint is very

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00:02:10,000 --> 00:02:14,000
simple, we can maybe solve for
one of the variables.

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00:02:14,000 --> 00:02:17,000
Maybe we can solve this
equation for one of the

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00:02:17,000 --> 00:02:21,000
variables, plug it back into f,
and then we have a usual

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00:02:21,000 --> 00:02:25,000
min/max problem that we have
seen how to do.

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00:02:25,000 --> 00:02:28,000
The problem is sometimes you
cannot actually solve for x,

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00:02:28,000 --> 00:02:31,000
y, z in here because this
condition is too complicated and

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00:02:31,000 --> 00:02:38,000
then we need a new method.
That is what we are going to do.

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00:02:38,000 --> 00:02:41,000
Why would we care about that?
Well, one example is actually

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00:02:41,000 --> 00:02:43,000
in physics.
Maybe you have seen in

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thermodynamics that you study
quantities about gases,

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00:02:47,000 --> 00:02:50,000
and those quantities that
involve pressure,

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00:02:50,000 --> 00:02:53,000
volume and temperature.
And pressure,

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volume and temperature are not
independent of each other.

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00:02:56,000 --> 00:02:59,000
I mean you know probably the
equation PV = NRT.

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00:02:59,000 --> 00:03:01,000
And, of course,
there you could actually solve

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00:03:01,000 --> 00:03:03,000
to express things in terms of
one or the other.

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00:03:03,000 --> 00:03:07,000
But sometimes it is more
convenient to keep all three

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00:03:07,000 --> 00:03:09,000
variables but treat them as
constrained.

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00:03:09,000 --> 00:03:19,000
It is just an example of a
situation where you might want

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00:03:19,000 --> 00:03:24,000
to do this.
Anyway, we will look mostly at

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particular examples,
but just to point out that this

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is useful when you study guesses
in physics.

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The first observation is we
cannot use our usual method of

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00:03:35,000 --> 00:03:36,000
looking for critical points of
f.

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00:03:36,000 --> 00:03:40,000
Because critical points of f
typically will not satisfy this

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condition and so won't be good
solutions.

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We need something else.
Let's look at an example,

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and we will see how that leads
us to the method.

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For example,
let's say that I want to find

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the point closest to the origin
-- -- on the hyperbola xy equals

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3 in the plane.
That means I have this

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hyperbola, and I am asking
myself what is the point on it

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that is the closest to the
origin?

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I mean we can solve this by
elementary geometry,

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we don't need actually Lagrange
multipliers,

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00:04:34,000 --> 00:04:38,000
but we are going to do it with
Lagrange multipliers because it

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00:04:38,000 --> 00:04:41,000
is a pretty good example.
What does it mean?

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00:04:41,000 --> 00:04:47,000
Well, it means that we want to
minimize distance to the origin.

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00:04:47,000 --> 00:04:49,000
What is the distance to the
origin?

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00:04:49,000 --> 00:04:53,000
If I have a point,
at coordinates (x,

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00:04:53,000 --> 00:04:58,000
y) and then the distance to the
origin is square root of x

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00:04:58,000 --> 00:05:02,000
squared plus y squared.
Well, do we really want to

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00:05:02,000 --> 00:05:05,000
minimize that or can we minimize
something easier?

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00:05:05,000 --> 00:05:06,000
Yeah.
Maybe we can minimize the

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00:05:06,000 --> 00:05:14,000
square of a distance.
Let's forget this guy and

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instead -- Actually,
we will minimize f of x,

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y equals x squared plus y
squared,

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00:05:27,000 --> 00:05:39,000
that looks better,
subject to the constraint xy =

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3.
And so we will call this thing

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g of x, y to illustrate the
general method.

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00:05:50,000 --> 00:05:58,000
Let's look at a picture.
Here you can see in yellow the

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00:05:58,000 --> 00:06:02,000
hyperbola xy equals three.
And we are going to look for

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the points that are the closest
to the origin.

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00:06:05,000 --> 00:06:08,000
What can we do?
Well, for example,

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00:06:08,000 --> 00:06:13,000
we can plot the function x
squared plus y squared,

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00:06:13,000 --> 00:06:17,000
function f.
That is the contour plot of f

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00:06:17,000 --> 00:06:21,000
with a hyperbola on top of it.
Now let's see what we can do

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with that.
Well, let's ask ourselves,

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for example,
if I look at points where f

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00:06:30,000 --> 00:06:34,000
equals 20 now.
I think I am at 20 but you

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00:06:34,000 --> 00:06:37,000
cannot really see it.
That is a circle with a point

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00:06:37,000 --> 00:06:41,000
whose distant square is 20.
Well, can I find a solution if

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00:06:41,000 --> 00:06:44,000
I am on the hyperbola?
Yes, there are four points at

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00:06:44,000 --> 00:06:46,000
this distance.
Can I do better?

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00:06:46,000 --> 00:06:49,000
Well, let's decrease for
distance.

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Yes, we can still find points
on the hyperbola and so on.

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00:06:52,000 --> 00:06:56,000
Except if we go too low then
there are no points on this

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circle anymore in the hyperbola.
If we decrease the value of f

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00:07:00,000 --> 00:07:03,000
that we want to look at that
will somehow limit value beyond

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00:07:03,000 --> 00:07:07,000
which we cannot go,
and that is the minimum of f.

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00:07:07,000 --> 00:07:13,000
We are trying to look for the
smallest value of f that will

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00:07:13,000 --> 00:07:17,000
actually be realized on the
hyperbola.

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00:07:17,000 --> 00:07:20,000
When does that happen?
Well, I have to backtrack a

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00:07:20,000 --> 00:07:23,000
little bit.
It seems like the limiting case

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is basically here.
It is when the circle is

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tangent to the hyperbola.
That is the smallest circle

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00:07:31,000 --> 00:07:37,000
that will hit the hyperbola.
If I take a larger value of f,

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I will have solutions.
If I take a smaller value of f,

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00:07:39,000 --> 00:07:41,000
I will not have any solutions
anymore.

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So, that is the situation that
we want to solve for.

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How do we find that minimum?
Well, a key observation that is

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valid on this picture,
and that actually remain true

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00:07:58,000 --> 00:08:03,000
in the completely general case,
is that when we have a minimum

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00:08:03,000 --> 00:08:09,000
the level curve of f is actually
tangent to our hyperbola.

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00:08:09,000 --> 00:08:15,000
It is tangent to the set of
points where x,

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00:08:15,000 --> 00:08:20,000
y equals three,
to the hyperbola.

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00:08:20,000 --> 00:08:32,000
Let's write that down.
We observe that at the minimum

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00:08:32,000 --> 00:08:49,000
the level curve of f is tangent
to the hyperbola.

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00:08:49,000 --> 00:08:53,000
Remember, the hyperbola is
given by the equal g equals

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00:08:53,000 --> 00:08:56,000
three, so it is a level curve of
g.

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00:08:56,000 --> 00:08:59,000
We have a level curve of f and
a level curve of g that are

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00:08:59,000 --> 00:09:03,000
tangent to each other.
And I claim that is going to be

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00:09:03,000 --> 00:09:07,000
the general situation that we
are interested in.

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00:09:07,000 --> 00:09:12,000
How do we try to solve for
points where this happens?

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00:09:28,000 --> 00:09:36,000
How do we find x,
y where the level curves of f

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00:09:36,000 --> 00:09:47,000
and g are tangent to each other?
Let's think for a second.

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00:09:47,000 --> 00:09:51,000
If the two level curves are
tangent to each other that means

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00:09:51,000 --> 00:09:57,000
they have the same tangent line.
That means that the normal

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00:09:57,000 --> 00:10:03,000
vectors should be parallel.
Let me maybe draw a picture

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00:10:03,000 --> 00:10:06,000
here.
This is the level curve maybe f

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00:10:06,000 --> 00:10:11,000
equals something.
And this is the level curve g

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00:10:11,000 --> 00:10:16,000
equals constant.
Here my constant is three.

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00:10:16,000 --> 00:10:20,000
Well, if I look for gradient
vectors, the gradient of f will

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00:10:20,000 --> 00:10:23,000
be perpendicular to the level
curve of f.

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00:10:23,000 --> 00:10:27,000
The gradient of g will be
perpendicular to the level curve

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00:10:27,000 --> 00:10:29,000
of g.
They don't have any reason to

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00:10:29,000 --> 00:10:32,000
be of the same size,
but they have to be parallel to

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00:10:32,000 --> 00:10:35,000
each other.
Of course, they could also be

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00:10:35,000 --> 00:10:38,000
parallel pointing in opposite
directions.

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00:10:38,000 --> 00:10:48,000
But the key point is that when
this happens the gradient of f

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00:10:48,000 --> 00:10:54,000
is parallel to the gradient of
g.

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00:10:54,000 --> 00:11:03,000
Well, let's check that.
Here is a point.

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00:11:03,000 --> 00:11:05,000
And I can plot the gradient of
f in blue.

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00:11:05,000 --> 00:11:08,000
The gradient of g in yellow.
And you see,

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00:11:08,000 --> 00:11:12,000
in most of these places,
somehow the two gradients are

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00:11:12,000 --> 00:11:14,000
not really parallel.
Actually, I should not be

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00:11:14,000 --> 00:11:17,000
looking at random points.
I should be looking only on the

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00:11:17,000 --> 00:11:19,000
hyperbola.
I want points on the hyperbola

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00:11:19,000 --> 00:11:22,000
where the two gradients are
parallel.

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00:11:22,000 --> 00:11:28,000
Well, when does that happen?
Well, it looks like it will

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00:11:28,000 --> 00:11:31,000
happen here.
When I am at a minimum,

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00:11:31,000 --> 00:11:34,000
the two gradient vectors are
parallel.

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00:11:34,000 --> 00:11:37,000
It is not really proof.
It is an example that seems to

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00:11:37,000 --> 00:11:43,000
be convincing.
So far things work pretty well.

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00:11:43,000 --> 00:11:46,000
How do we decide if two vectors
are parallel?

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00:11:46,000 --> 00:11:50,000
Well, they are parallel when
they are proportional to each

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00:11:50,000 --> 00:11:54,000
other.
You can write one of them as a

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00:11:54,000 --> 00:12:02,000
constant times the other one,
and that constant usually one

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00:12:02,000 --> 00:12:07,000
uses the Greek letter lambda.
I don't know if you have seen

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00:12:07,000 --> 00:12:10,000
it before.
It is the Greek letter for L.

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00:12:10,000 --> 00:12:15,000
And probably,
I am sure, it is somebody's

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00:12:15,000 --> 00:12:22,000
idea of paying tribute to
Lagrange by putting an L in

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00:12:22,000 --> 00:12:25,000
there.
Lambda is just a constant.

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00:12:25,000 --> 00:12:31,000
And we are looking for a scalar
lambda and points x and y where

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00:12:31,000 --> 00:12:33,000
this holds.
In fact,

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00:12:33,000 --> 00:12:37,000
what we are doing is replacing
min/max problems in two

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00:12:37,000 --> 00:12:41,000
variables with a constraint
between them by a set of

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00:12:41,000 --> 00:12:47,000
equations involving,
you will see, three variables.

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00:12:47,000 --> 00:12:54,000
We had min/max with two
variables x, y,

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00:12:54,000 --> 00:13:00,000
but no independent.
We had a constraint g of x,

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00:13:00,000 --> 00:13:06,000
y equals constant.
And that becomes something new.

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00:13:06,000 --> 00:13:12,000
That becomes a system of
equations where we have to

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00:13:12,000 --> 00:13:19,000
solve, well, let's write down
what it means for gradient f to

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00:13:19,000 --> 00:13:26,000
be proportional to gradient g.
That means that f sub x should

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00:13:26,000 --> 00:13:32,000
be lambda times g sub x,
and f sub y should be lambda

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00:13:32,000 --> 00:13:36,000
times g sub y.
Because the gradient vectors

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00:13:36,000 --> 00:13:39,000
here are f sub x,
f sub y and g sub x,

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00:13:39,000 --> 00:13:43,000
g sub y.
If you have a third variable z

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00:13:43,000 --> 00:13:49,000
then you have also an equation f
sub z equals lambda g sub z.

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00:13:49,000 --> 00:13:53,000
Now, let's see.
How many unknowns do we have in

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00:13:53,000 --> 00:13:55,000
these equations?
Well, there is x,

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00:13:55,000 --> 00:14:01,000
there is y and there is lambda.
We have three unknowns and have

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00:14:01,000 --> 00:14:06,000
only two equations.
Something is missing.

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00:14:06,000 --> 00:14:10,000
Well, I mean x and y are not
actually independent.

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00:14:10,000 --> 00:14:14,000
They are related by the
equation g of x,

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00:14:14,000 --> 00:14:21,000
y equals c, so we need to add
the constraint g equals c.

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00:14:21,000 --> 00:14:26,000
And now we have three equations
involving three variables.

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00:14:26,000 --> 00:14:39,000
Let's see how that works.
Here remember we have f equals

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00:14:39,000 --> 00:14:45,000
x squared y squared and g = xy.
What is f sub x?

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00:14:45,000 --> 00:14:52,000
It is going to be 2x equals
lambda times,

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00:14:52,000 --> 00:14:55,000
what is g sub x,
y.

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00:14:55,000 --> 00:14:59,000
Maybe I should write here f sub
x equals lambda g sub x just to

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00:14:59,000 --> 00:15:03,000
remind you.
Then we have f sub y equals

192
00:15:03,000 --> 00:15:10,000
lambda g sub y.
F sub y is 2y equals lambda

193
00:15:10,000 --> 00:15:18,000
times g sub y is x.
And then our third equation g

194
00:15:18,000 --> 00:15:22,000
equals c becomes xy equals
three.

195
00:15:22,000 --> 00:15:26,000
So, that is what you would have
to solve.

196
00:15:26,000 --> 00:15:33,000
Any questions at this point?
No.

197
00:15:33,000 --> 00:15:44,000
Yes?
How do I know the direction of

198
00:15:44,000 --> 00:15:47,000
a gradient?
Do you mean how do I know that

199
00:15:47,000 --> 00:15:50,000
it is perpendicular to a level
curve?

200
00:15:50,000 --> 00:15:54,000
Oh, how do I know if it points
in that direction on the

201
00:15:54,000 --> 00:15:56,000
opposite one?
Well, that depends.

202
00:15:56,000 --> 00:15:59,000
I mean we'd seen in last time,
but the gradient is

203
00:15:59,000 --> 00:16:02,000
perpendicular to the level and
points towards higher values of

204
00:16:02,000 --> 00:16:05,000
a function.
So it could be -- Wait.

205
00:16:05,000 --> 00:16:08,000
What did I have?
It could be that my gradient

206
00:16:08,000 --> 00:16:11,000
vectors up there actually point
in opposite directions.

207
00:16:11,000 --> 00:16:15,000
It doesn't matter to me because
it will still look the same in

208
00:16:15,000 --> 00:16:18,000
terms of the equation,
just lambda will be positive or

209
00:16:18,000 --> 00:16:22,000
negative, depending on the case.
I can handle both situations.

210
00:16:22,000 --> 00:16:30,000
It's not a problem.
I can allow lambda to be

211
00:16:30,000 --> 00:16:34,000
positive or negative.
Well, in this example,

212
00:16:34,000 --> 00:16:35,000
it looks like lambda will be
positive.

213
00:16:35,000 --> 00:16:38,000
If you look at the picture on
the plot.

214
00:16:38,000 --> 00:16:48,000
Yes?
Well, because actually they are

215
00:16:48,000 --> 00:16:51,000
not equal to each other.
If you look at this point where

216
00:16:51,000 --> 00:16:55,000
the hyperbola and the circle
touch each other,

217
00:16:55,000 --> 00:16:58,000
first of all,
I don't know which circle I am

218
00:16:58,000 --> 00:17:01,000
going to look at.
I am trying to solve,

219
00:17:01,000 --> 00:17:04,000
actually, for the radius of the
circle.

220
00:17:04,000 --> 00:17:07,000
I am trying to find what the
minimum value of f is.

221
00:17:07,000 --> 00:17:10,000
And, second,
at that point,

222
00:17:10,000 --> 00:17:14,000
the value of f and the value of
g are not equal.

223
00:17:14,000 --> 00:17:17,000
g is equal to three because I
want the hyperbola x equals

224
00:17:17,000 --> 00:17:19,000
three.
The value of f will be the

225
00:17:19,000 --> 00:17:22,000
square of a distance,
whatever that is.

226
00:17:22,000 --> 00:17:27,000
I think it will end up being 6,
but we will see.

227
00:17:27,000 --> 00:17:29,000
So, you cannot really set them
equal because you don't know

228
00:17:29,000 --> 00:17:45,000
what f is equal to in advance.
Yes?

229
00:17:45,000 --> 00:17:49,000
Not quite.
Actually, here I am just using

230
00:17:49,000 --> 00:17:52,000
this idea of finding a point
closest to the origin to

231
00:17:52,000 --> 00:17:55,000
illustrate an example of a
min/max problem.

232
00:17:55,000 --> 00:17:59,000
The general problem we are
trying to solve is minimize f

233
00:17:59,000 --> 00:18:03,000
subject to g equals constant.
And what we are going to do for

234
00:18:03,000 --> 00:18:07,000
that is we are really going to
say instead let's look at places

235
00:18:07,000 --> 00:18:10,000
where gradient f and gradient g
are parallel to each other and

236
00:18:10,000 --> 00:18:14,000
solve for equations of that.
I think we completely lose the

237
00:18:14,000 --> 00:18:19,000
notion of closest point if we
just look at these equations.

238
00:18:19,000 --> 00:18:21,000
We don't really say anything
about closest points anymore.

239
00:18:21,000 --> 00:18:24,000
Of course, that is what they
mean in the end.

240
00:18:24,000 --> 00:18:28,000
But, in the general setting,
there is no closest point

241
00:18:28,000 --> 00:18:31,000
involved anymore.
OK.

242
00:18:31,000 --> 00:18:40,000
Yes?
Yes.

243
00:18:40,000 --> 00:18:43,000
It is always going to be the
case that,

244
00:18:43,000 --> 00:18:46,000
at the minimum,
or at the maximum of a function

245
00:18:46,000 --> 00:18:49,000
subject to a constraint,
the level curves of f and the

246
00:18:49,000 --> 00:18:52,000
level curves of g will be
tangent to each other.

247
00:18:52,000 --> 00:18:54,000
That is the basis for this
method.

248
00:18:54,000 --> 00:19:00,000
I am going to justify that soon.
It could be minimum or maximum.

249
00:19:00,000 --> 00:19:02,000
In three-dimensions it could
even be a saddle point.

250
00:19:02,000 --> 00:19:03,000
And, in fact,
I should say in advance,

251
00:19:03,000 --> 00:19:06,000
this method will not tell us
whether it is a minimum or a

252
00:19:06,000 --> 00:19:08,000
maximum.
We do not have any way of

253
00:19:08,000 --> 00:19:10,000
knowing, except for testing
values.

254
00:19:10,000 --> 00:19:13,000
We cannot use second derivative
tests or anything like that.

255
00:19:13,000 --> 00:19:21,000
I will get back to that.
Yes?

256
00:19:21,000 --> 00:19:23,000
Yes.
Here you can set y equals to

257
00:19:23,000 --> 00:19:26,000
favor x.
Then you can minimize x squared

258
00:19:26,000 --> 00:19:30,000
plus nine over x squared.
In general, if I am trying to

259
00:19:30,000 --> 00:19:33,000
solve a more complicated
problem, I might not be able to

260
00:19:33,000 --> 00:19:35,000
solve.
I am doing an example where,

261
00:19:35,000 --> 00:19:38,000
indeed, here you could solve
and remove one variable,

262
00:19:38,000 --> 00:19:41,000
but you cannot always do that.
And this method will still work.

263
00:19:41,000 --> 00:19:47,000
The other one won't.
OK.

264
00:19:47,000 --> 00:19:53,000
I don't see any other questions.
Are there any other questions?

265
00:19:53,000 --> 00:19:56,000
No.
OK.

266
00:19:56,000 --> 00:20:02,000
I see a lot of students
stretching and so on,

267
00:20:02,000 --> 00:20:08,000
so it is very confusing for me.
How do we solve these equations?

268
00:20:08,000 --> 00:20:14,000
Well, the answer is in general
we might be in deep trouble.

269
00:20:14,000 --> 00:20:18,000
There is no general method for
solving the equations that you

270
00:20:18,000 --> 00:20:21,000
get from this method.
You just have to think about

271
00:20:21,000 --> 00:20:25,000
them.
Sometimes it will be very easy.

272
00:20:25,000 --> 00:20:28,000
Sometimes it will be so hard
that you cannot actually do it

273
00:20:28,000 --> 00:20:31,000
without the computer.
Sometimes it will be just hard

274
00:20:31,000 --> 00:20:33,000
enough to be on Part B of this
week's problem set.

275
00:20:50,000 --> 00:20:56,000
I claim in this case we can
actually do it without so much

276
00:20:56,000 --> 00:21:03,000
trouble, because actually we can
think of this as a two by two

277
00:21:03,000 --> 00:21:10,000
linear system in x and y.
Well, let me do something.

278
00:21:10,000 --> 00:21:18,000
Let me rewrite the first two
equations as 2x - lambda y = 0.

279
00:21:18,000 --> 00:21:30,000
And lambda x - 2y = 0.
And xy = 3.

280
00:21:30,000 --> 00:21:36,000
That is what we want to solve.
Well, I can put this into

281
00:21:36,000 --> 00:21:41,000
matrix form.
Two minus lambda,

282
00:21:41,000 --> 00:21:48,000
lambda minus two times x,
y equals 0,0.

283
00:21:48,000 --> 00:21:52,000
Now, how do I solve a linear
system matrix times x,

284
00:21:52,000 --> 00:21:54,000
y equals zero?
Well, I always have an obvious

285
00:21:54,000 --> 00:21:56,000
solution.
X and y both equal to zero.

286
00:21:56,000 --> 00:22:02,000
Is that a good solution?
No, because zero times zero is

287
00:22:02,000 --> 00:22:07,000
not three.
We want another solution,

288
00:22:07,000 --> 00:22:14,000
the trivial solution.
0,0 does not solve the

289
00:22:14,000 --> 00:22:20,000
constraint equation xy equals
three, so we want another

290
00:22:20,000 --> 00:22:24,000
solution.
When do we have another

291
00:22:24,000 --> 00:22:29,000
solution?
Well, when the determinant of a

292
00:22:29,000 --> 00:22:37,000
matrix is zero.
We have other solutions that

293
00:22:37,000 --> 00:22:46,000
exist only if determinant of a
matrix is zero.

294
00:22:46,000 --> 00:23:01,000
M is this guy.
Let's compute the determinant.

295
00:23:01,000 --> 00:23:08,000
Well, that seems to be negative
four plus lambda squared.

296
00:23:08,000 --> 00:23:15,000
That is zero exactly when
lambda squared equals four,

297
00:23:15,000 --> 00:23:20,000
which is lambda is plus or
minus two.

298
00:23:20,000 --> 00:23:25,000
Already you see here it is a
the level of difficulty that is

299
00:23:25,000 --> 00:23:30,000
a little bit much for an exam
but perfectly fine for a problem

300
00:23:30,000 --> 00:23:33,000
set or for a beautiful lecture
like this one.

301
00:23:33,000 --> 00:23:37,000
How do we deal with -- Well,
we have two cases to look at.

302
00:23:37,000 --> 00:23:40,000
Lambda equals two or lambda
equals minus two.

303
00:23:40,000 --> 00:23:43,000
Let's start with lambda equals
two.

304
00:23:43,000 --> 00:23:47,000
If I set lambda equals two,
what does this equation become?

305
00:23:47,000 --> 00:23:53,000
Well, it becomes x equals y.
This one becomes y equals x.

306
00:23:53,000 --> 00:23:57,000
Well, they seem to be the same.
x equals y.

307
00:23:57,000 --> 00:24:01,000
And then the equation xy equals
three becomes,

308
00:24:01,000 --> 00:24:06,000
well, x squared equals three.
I have two solutions.

309
00:24:06,000 --> 00:24:15,000
One is x equals root three and,
therefore, y equals root three

310
00:24:15,000 --> 00:24:23,000
as well, or negative root three
and negative root three.

311
00:24:23,000 --> 00:24:26,000
Let's look at the other case.
If I set lambda equal to

312
00:24:26,000 --> 00:24:30,000
negative two then I get 2x
equals negative 2y.

313
00:24:30,000 --> 00:24:37,000
That means x equals negative y.
The second one,

314
00:24:37,000 --> 00:24:40,000
2y equals negative 2x.
That is y equals negative x.

315
00:24:40,000 --> 00:24:45,000
Well, that is the same thing.
And xy equals three becomes

316
00:24:45,000 --> 00:24:51,000
negative x squared equals three.
Can we solve that?

317
00:24:51,000 --> 00:24:58,000
No.
There are no solutions here.

318
00:24:58,000 --> 00:25:03,000
Now we have two candidate
points which are these two

319
00:25:03,000 --> 00:25:07,000
points, root three,
root three or negative root

320
00:25:07,000 --> 00:25:13,000
three, negative root three.
OK.

321
00:25:13,000 --> 00:25:16,000
Let's actually look at what we
have here.

322
00:25:16,000 --> 00:25:20,000
Maybe you cannot read the
coordinates, but the point that

323
00:25:20,000 --> 00:25:23,000
I have here is indeed root
three, root three.

324
00:25:23,000 --> 00:25:26,000
How do we see that lambda
equals two?

325
00:25:26,000 --> 00:25:29,000
Well, if you look at this
picture, the gradient of f,

326
00:25:29,000 --> 00:25:32,000
that is the blue vector,
is indeed twice the yellow

327
00:25:32,000 --> 00:25:36,000
vector, gradient g.
That is where you read the

328
00:25:36,000 --> 00:25:41,000
value of lambda.
And we have the other solution

329
00:25:41,000 --> 00:25:45,000
which is somewhere here.
Negative root three,

330
00:25:45,000 --> 00:25:48,000
negative root there.
And there, again,

331
00:25:48,000 --> 00:25:51,000
lambda equals two.
The two vectors are

332
00:25:51,000 --> 00:25:59,000
proportional by a factor of two.
Yes?

333
00:25:59,000 --> 00:26:01,000
No, solutions are not quite
guaranteed to be absolute minima

334
00:26:01,000 --> 00:26:03,000
or maxima.
They are guaranteed to be

335
00:26:03,000 --> 00:26:06,000
somehow critical points end of a
constraint.

336
00:26:06,000 --> 00:26:09,000
That means if you were able to
solve and eliminate the variable

337
00:26:09,000 --> 00:26:12,000
that would be a critical point.
When you have the same problem,

338
00:26:12,000 --> 00:26:14,000
as we have critical points,
are they maxima or minima?

339
00:26:14,000 --> 00:26:22,000
And the answer is,
well, we won't know until we

340
00:26:22,000 --> 00:26:28,000
check.
More questions?

341
00:26:28,000 --> 00:26:32,000
No.
Yes?

342
00:26:32,000 --> 00:26:36,000
What is a Lagrange multiplier?
Well, it is this number lambda

343
00:26:36,000 --> 00:26:39,000
that is called the multiplier
here.

344
00:26:39,000 --> 00:26:44,000
It is a multiplier because it
is what you have to multiply

345
00:26:44,000 --> 00:26:48,000
gradient of g by to get gradient
of f.

346
00:26:48,000 --> 00:26:49,000
It multiplies.

347
00:27:04,000 --> 00:27:11,000
Let's try to see why is this
method valid?

348
00:27:11,000 --> 00:27:18,000
Because so far I have shown you
pictures and have said see they

349
00:27:18,000 --> 00:27:23,000
are tangent.
But why is it that they have to

350
00:27:23,000 --> 00:27:28,000
be tangent in general?
Let's think about it.

351
00:27:28,000 --> 00:27:37,000
Let's say that we are at
constrained min or max.

352
00:27:37,000 --> 00:27:42,000
What that means is that if I
move on the level g equals

353
00:27:42,000 --> 00:27:46,000
constant then the value of f
should only increase or only

354
00:27:46,000 --> 00:27:49,000
decrease.
But it means,

355
00:27:49,000 --> 00:27:53,000
in particular,
to first order it will not

356
00:27:53,000 --> 00:27:56,000
change.
At an unconstrained min or max,

357
00:27:56,000 --> 00:27:59,000
partial derivatives are zero.
In this case,

358
00:27:59,000 --> 00:28:02,000
derivatives are zero only in
the allowed directions.

359
00:28:02,000 --> 00:28:09,000
And the allowed directions are
those that stay on the levels of

360
00:28:09,000 --> 00:28:21,000
this g equals constant.
In any direction along the

361
00:28:21,000 --> 00:28:40,000
level set g = c the rate of
change of f must be zero.

362
00:28:40,000 --> 00:28:44,000
That is what happens at minima
or maxima.

363
00:28:44,000 --> 00:28:49,000
Except here,
of course, we look only at the

364
00:28:49,000 --> 00:28:54,000
allowed directions.
Let's say the same thing in

365
00:28:54,000 --> 00:28:57,000
terms of directional
derivatives.

366
00:29:23,000 --> 00:29:35,000
That means for any direction
that is tangent to the

367
00:29:35,000 --> 00:29:49,000
constraint level g equal c,
we must have df over ds in the

368
00:29:49,000 --> 00:30:00,000
direction of u equals zero.
I will draw a picture.

369
00:30:00,000 --> 00:30:05,000
Let's say now I am in three
variables just to give you

370
00:30:05,000 --> 00:30:09,000
different examples.
Here I have a level surface g

371
00:30:09,000 --> 00:30:11,000
equals c.
I am at my point.

372
00:30:11,000 --> 00:30:18,000
And if I move in any direction
that is on the level surface,

373
00:30:18,000 --> 00:30:24,000
so I move in the direction u
tangent to the level surface,

374
00:30:24,000 --> 00:30:32,000
then the rate of change of f in
that direction should be zero.

375
00:30:32,000 --> 00:30:34,000
Now, remember what the formula
is for this guy.

376
00:30:34,000 --> 00:30:44,000
Well, we have seen that this
guy is actually radiant f dot u.

377
00:30:44,000 --> 00:30:58,000
That means any such vector u
must be perpendicular to the

378
00:30:58,000 --> 00:31:05,000
gradient of f.
That means that the gradient of

379
00:31:05,000 --> 00:31:10,000
f should be perpendicular to
anything that is tangent to this

380
00:31:10,000 --> 00:31:12,000
level.
That means the gradient of f

381
00:31:12,000 --> 00:31:16,000
should be perpendicular to the
level set.

382
00:31:16,000 --> 00:31:17,000
That is what we have shown.

383
00:31:37,000 --> 00:31:40,000
But we know another vector that
is also perpendicular to the

384
00:31:40,000 --> 00:31:57,000
level set of g.
That is the gradient of g.

385
00:31:57,000 --> 00:32:02,000
We conclude that the gradient
of f must be parallel to the

386
00:32:02,000 --> 00:32:07,000
gradient of g because both are
perpendicular to the level set

387
00:32:07,000 --> 00:32:09,000
of g.
I see confused faces,

388
00:32:09,000 --> 00:32:13,000
so let me try to tell you again
where that comes from.

389
00:32:13,000 --> 00:32:16,000
We said if we had a constrained
minimum or maximum,

390
00:32:16,000 --> 00:32:19,000
if we move in the level set of
g, f doesn't change.

391
00:32:19,000 --> 00:32:20,000
Well, it doesn't change to
first order.

392
00:32:20,000 --> 00:32:24,000
It is the same idea as when you
are looking for a minimum you

393
00:32:24,000 --> 00:32:26,000
set the derivative equal to
zero.

394
00:32:26,000 --> 00:32:31,000
So the derivative in any
direction, tangent to g equals

395
00:32:31,000 --> 00:32:34,000
c, should be the directional
derivative of f,

396
00:32:34,000 --> 00:32:38,000
in any such direction,
should be zero.

397
00:32:38,000 --> 00:32:43,000
That is what we mean by
critical point of f.

398
00:32:43,000 --> 00:32:48,000
And so that means that any
vector u, any unit vector

399
00:32:48,000 --> 00:32:55,000
tangent to the level set of g is
going to be perpendicular to the

400
00:32:55,000 --> 00:33:00,000
gradient of f.
That means that the gradient of

401
00:33:00,000 --> 00:33:04,000
f is perpendicular to the level
set of g.

402
00:33:04,000 --> 00:33:06,000
If you want,
that means the level sets of f

403
00:33:06,000 --> 00:33:10,000
and g are tangent to each other.
That is justifying what we have

404
00:33:10,000 --> 00:33:15,000
observed in the picture that the
two level sets have to be

405
00:33:15,000 --> 00:33:20,000
tangent to each other at the
prime minimum or maximum.

406
00:33:20,000 --> 00:33:23,000
Does that make a little bit of
sense?

407
00:33:23,000 --> 00:33:28,000
Kind of.
I see at least a few faces

408
00:33:28,000 --> 00:33:35,000
nodding so I take that to be a
positive answer.

409
00:33:35,000 --> 00:33:39,000
Since I have been asked by
several of you,

410
00:33:39,000 --> 00:33:43,000
how do I know if it is a
maximum or a minimum?

411
00:33:43,000 --> 00:33:57,000
Well, warning,
the method doesn't tell whether

412
00:33:57,000 --> 00:34:09,000
a solution is a minimum or a
maximum.

413
00:34:09,000 --> 00:34:13,000
How do we do it?
Well, more bad news.

414
00:34:13,000 --> 00:34:26,000
We cannot use the second
derivative test.

415
00:34:26,000 --> 00:34:30,000
And the reason for that is that
we care actually only about

416
00:34:30,000 --> 00:34:34,000
these specific directions that
are tangent to variable of g.

417
00:34:34,000 --> 00:34:39,000
And we don't want to bother to
try to define directional second

418
00:34:39,000 --> 00:34:42,000
derivatives.
Not to mention that actually it

419
00:34:42,000 --> 00:34:45,000
wouldn't work.
There is a criterion but it is

420
00:34:45,000 --> 00:34:49,000
much more complicated than that.
Basically, the answer for us is

421
00:34:49,000 --> 00:34:52,000
that we don't have a second
derivative test in this

422
00:34:52,000 --> 00:34:54,000
situation.
What are we left with?

423
00:34:54,000 --> 00:34:57,000
Well, we are just left with
comparing values.

424
00:34:57,000 --> 00:35:00,000
Say that in this problem you
found a point where f equals

425
00:35:00,000 --> 00:35:04,000
three, a point where f equals
nine, a point where f equals 15.

426
00:35:04,000 --> 00:35:08,000
Well, then probably the minimum
is the point where f equals

427
00:35:08,000 --> 00:35:12,000
three and the maximum is 15.
Actually, in this case,

428
00:35:12,000 --> 00:35:17,000
where we found minima,
these two points are tied for

429
00:35:17,000 --> 00:35:19,000
minimum.
What about the maximum?

430
00:35:19,000 --> 00:35:22,000
What is the maximum of f on the
hyperbola?

431
00:35:22,000 --> 00:35:25,000
Well, it is infinity because
the point can go as far as you

432
00:35:25,000 --> 00:35:29,000
want from the origin.
But the general idea is if we

433
00:35:29,000 --> 00:35:35,000
have a good reason to believe
that there should be a minimum,

434
00:35:35,000 --> 00:35:38,000
and it's not like at infinity
or something weird like that,

435
00:35:38,000 --> 00:35:42,000
then the minimum will be a
solution of the Lagrange

436
00:35:42,000 --> 00:35:46,000
multiplier equations.
We just look for all the

437
00:35:46,000 --> 00:35:51,000
solutions and then we choose the
one that gives us the lowest

438
00:35:51,000 --> 00:35:55,000
value.
Is that good enough?

439
00:35:55,000 --> 00:35:57,000
Let me actually write that down.

440
00:36:23,000 --> 00:36:35,000
To find the minimum or the
maximum, we compare values of f

441
00:36:35,000 --> 00:36:46,000
at the various solutions -- --
to Lagrange multiplier

442
00:36:46,000 --> 00:36:49,000
equations.

443
00:37:08,000 --> 00:37:11,000
I should say also that
sometimes you can just conclude

444
00:37:11,000 --> 00:37:14,000
by thinking geometrically.
In this case,

445
00:37:14,000 --> 00:37:18,000
when it is asking you which
point is closest to the origin

446
00:37:18,000 --> 00:37:23,000
you can just see that your
answer is the correct one.

447
00:37:23,000 --> 00:37:32,000
Let's do an advanced example.
Advanced means that -- Well,

448
00:37:32,000 --> 00:37:37,000
this one I didn't actually dare
to put on top of the other

449
00:37:37,000 --> 00:37:48,000
problem sets.
Instead, I am going to do it.

450
00:37:48,000 --> 00:37:51,000
What is this going to be about?
We are going to look for a

451
00:37:51,000 --> 00:38:03,000
surface minimizing pyramid.
Let's say that we want to build

452
00:38:03,000 --> 00:38:19,000
a pyramid with a given
triangular base -- -- and a

453
00:38:19,000 --> 00:38:28,000
given volume.
Say that I have maybe in the x,

454
00:38:28,000 --> 00:38:33,000
y plane I am giving you some
triangle.

455
00:38:33,000 --> 00:38:40,000
And I am going to try to build
a pyramid.

456
00:38:40,000 --> 00:38:48,000
Of course, I can choose where
to put the top of a pyramid.

457
00:38:48,000 --> 00:38:53,000
This guy will end up being
behind now.

458
00:38:53,000 --> 00:39:09,000
And the constraint and the goal
is to minimize the total surface

459
00:39:09,000 --> 00:39:13,000
area.
The first time I taught this

460
00:39:13,000 --> 00:39:15,000
class, it was a few years ago,
was just before they built the

461
00:39:15,000 --> 00:39:17,000
Stata Center.
And then I used to motivate

462
00:39:17,000 --> 00:39:20,000
this problem by saying Frank
Gehry has gone crazy and has

463
00:39:20,000 --> 00:39:23,000
been given a triangular plot of
land he wants to put a pyramid.

464
00:39:23,000 --> 00:39:26,000
There needs to be the right
amount of volume so that you can

465
00:39:26,000 --> 00:39:28,000
put all the offices in there.
And he wants it to be,

466
00:39:28,000 --> 00:39:31,000
actually, covered in solid
gold.

467
00:39:31,000 --> 00:39:34,000
And because that is expensive,
the administration wants him to

468
00:39:34,000 --> 00:39:38,000
cut the costs a bit.
And so you have to minimize the

469
00:39:38,000 --> 00:39:42,000
total size so that it doesn't
cost too much.

470
00:39:42,000 --> 00:39:45,000
We will see if MIT comes up
with a triangular pyramid

471
00:39:45,000 --> 00:39:48,000
building.
Hopefully not.

472
00:39:48,000 --> 00:39:58,000
It could be our next dorm,
you never know.

473
00:39:58,000 --> 00:40:01,000
Anyway, it is a fine geometry
problem.

474
00:40:01,000 --> 00:40:07,000
Let's try to think about how we
can do this.

475
00:40:07,000 --> 00:40:10,000
The natural way to think about
it would be -- Well,

476
00:40:10,000 --> 00:40:11,000
what do we have to look for
first?

477
00:40:11,000 --> 00:40:18,000
We have to look for the
position of that top point.

478
00:40:18,000 --> 00:40:29,000
Remember we know that the
volume of a pyramid is one-third

479
00:40:29,000 --> 00:40:37,000
the area of base times height.
In fact, fixing the volume,

480
00:40:37,000 --> 00:40:39,000
knowing that we have fixed the
area of a base,

481
00:40:39,000 --> 00:40:43,000
means that we are fixing the
height of the pyramid.

482
00:40:43,000 --> 00:40:47,000
The height is completely fixed.
What we have to choose just is

483
00:40:47,000 --> 00:40:52,000
where do we put that top point?
Do we put it smack in the

484
00:40:52,000 --> 00:40:58,000
middle of a triangle or to a
side or even anywhere we want?

485
00:40:58,000 --> 00:41:15,000
Its z coordinate is fixed.
Let's call h the height.

486
00:41:15,000 --> 00:41:20,000
What we could do is something
like this.

487
00:41:20,000 --> 00:41:24,000
We say we have three points of
a base.

488
00:41:24,000 --> 00:41:32,000
Let's call them p1 at (x1,
y1,0); p2 at (x2,

489
00:41:32,000 --> 00:41:36,000
y2,0); p3 at (x3,
y3,0).

490
00:41:36,000 --> 00:41:40,000
This point p is the unknown
point at (x, y,

491
00:41:40,000 --> 00:41:42,000
h).
We know the height.

492
00:41:42,000 --> 00:41:46,000
And then we want to minimize
the sum of the areas of these

493
00:41:46,000 --> 00:41:50,000
three triangles.
One here, one here and one at

494
00:41:50,000 --> 00:41:53,000
the back.
And areas of triangles we know

495
00:41:53,000 --> 00:41:57,000
how to express by using length
of cross-product.

496
00:41:57,000 --> 00:42:00,000
It becomes a function of x and
y.

497
00:42:00,000 --> 00:42:04,000
And you can try to minimize it.
Actually, it doesn't quite work.

498
00:42:04,000 --> 00:42:05,000
The formulas are just too
complicated.

499
00:42:05,000 --> 00:42:14,000
You will never get there.
What happens is actually maybe

500
00:42:14,000 --> 00:42:18,000
we need better coordinates.
Why do we need better

501
00:42:18,000 --> 00:42:21,000
coordinates?
That is because the geometry is

502
00:42:21,000 --> 00:42:24,000
kind of difficult to do if you
use x, y coordinates.

503
00:42:24,000 --> 00:42:28,000
I mean formula for
cross-product is fine,

504
00:42:28,000 --> 00:42:33,000
but then the length of the
vector will be annoying and just

505
00:42:33,000 --> 00:42:37,000
doesn't look good.
Instead, let's think about it

506
00:42:37,000 --> 00:42:38,000
differently.

507
00:42:54,000 --> 00:43:01,000
I claim if we do it this way
and we express the area as a

508
00:43:01,000 --> 00:43:06,000
function of x,
y, well, actually we can't

509
00:43:06,000 --> 00:43:13,000
solve for a minimum.
Here is another way to do it.

510
00:43:13,000 --> 00:43:17,000
Well, what has worked pretty
well for us so far is this

511
00:43:17,000 --> 00:43:19,000
geometric idea of base times
height.

512
00:43:19,000 --> 00:43:29,000
So let's think in terms of the
heights of side triangles.

513
00:43:29,000 --> 00:43:37,000
I am going to use the height of
these things.

514
00:43:37,000 --> 00:43:43,000
And I am going to say that the
area will be the sum of three

515
00:43:43,000 --> 00:43:48,000
terms, which are three bases
times three heights.

516
00:43:48,000 --> 00:43:53,000
Let's give names to these
quantities.

517
00:43:53,000 --> 00:43:58,000
Actually, for that it is going
to be good to have the point in

518
00:43:58,000 --> 00:44:01,000
the xy plane that lives directly
below p.

519
00:44:01,000 --> 00:44:08,000
Let's call it q.
P is the point that coordinates

520
00:44:08,000 --> 00:44:13,000
x, y, h.
And let's call q the point that

521
00:44:13,000 --> 00:44:19,000
is just below it and so it'
coordinates are x,

522
00:44:19,000 --> 00:44:22,000
y, 0.
Let's see.

523
00:44:22,000 --> 00:44:34,000
Let me draw a map of this thing.
p1, p2, p3 and I have my point

524
00:44:34,000 --> 00:44:37,000
q in the middle.
Let's see.

525
00:44:37,000 --> 00:44:40,000
To know these areas,
I need to know the base.

526
00:44:40,000 --> 00:44:44,000
Well, the base I can decide
that I know it because it is

527
00:44:44,000 --> 00:44:48,000
part of my given data.
I know the sides of this

528
00:44:48,000 --> 00:44:53,000
triangle.
Let me call the lengths a1,

529
00:44:53,000 --> 00:44:56,000
a2, a3.
I also need to know the height,

530
00:44:56,000 --> 00:44:58,000
so I need to know these
lengths.

531
00:44:58,000 --> 00:45:01,000
How do I know these lengths?
Well, its distance in space,

532
00:45:01,000 --> 00:45:03,000
but it is a little bit
annoying.

533
00:45:03,000 --> 00:45:10,000
But maybe I can reduce it to a
distance in the plane by looking

534
00:45:10,000 --> 00:45:17,000
instead at this distance here.
Let me give names to the

535
00:45:17,000 --> 00:45:24,000
distances from q to the sides.
Let's call u1,

536
00:45:24,000 --> 00:45:35,000
u2, u3 the distances from q to
the sides.

537
00:45:47,000 --> 00:45:49,000
Well, now I can claim I can
find, actually,

538
00:45:49,000 --> 00:45:53,000
sorry.
I need to draw one more thing.

539
00:45:53,000 --> 00:45:57,000
I claim I have a nice formula
for the area,

540
00:45:57,000 --> 00:46:01,000
because this is vertical and
this is horizontal so this

541
00:46:01,000 --> 00:46:05,000
length here is u3,
this length here is h.

542
00:46:05,000 --> 00:46:13,000
So what is this length here?
It is the square root of u3

543
00:46:13,000 --> 00:46:17,000
squared plus h squared.
And similarly for these other

544
00:46:17,000 --> 00:46:23,000
guys.
They are square roots of a u

545
00:46:23,000 --> 00:46:31,000
squared plus h squared.
The heights of the faces are

546
00:46:31,000 --> 00:46:36,000
square root of u1 squared times
h squared.

547
00:46:36,000 --> 00:46:43,000
And similarly with u2 and u3.
So the total side area is going

548
00:46:43,000 --> 00:46:47,000
to be the area of the first
faces,

549
00:46:47,000 --> 00:46:58,000
one-half of base times height,
plus one-half of a base times a

550
00:46:58,000 --> 00:47:06,000
height plus one-half of the
third one.

551
00:47:06,000 --> 00:47:09,000
It doesn't look so much better.
But, trust me,

552
00:47:09,000 --> 00:47:15,000
it will get better.
Now, that is a function of

553
00:47:15,000 --> 00:47:19,000
three variables,
u1, u2, u3.

554
00:47:19,000 --> 00:47:22,000
And how do we relate u1,
u2, u3 to each other?

555
00:47:22,000 --> 00:47:25,000
They are probably not
independent.

556
00:47:25,000 --> 00:47:32,000
Well, let's cut this triangle
here into three pieces like

557
00:47:32,000 --> 00:47:35,000
that.
Then each piece has side --

558
00:47:35,000 --> 00:47:40,000
Well, let's look at it the piece
of the bottom.

559
00:47:40,000 --> 00:47:50,000
It has base a3, height u3.
Cutting base into three tells

560
00:47:50,000 --> 00:47:57,000
you that the area of a base is
one-half of a1,

561
00:47:57,000 --> 00:48:04,000
u1 plus one-half of a2,
u2 plus one-half of a3,

562
00:48:04,000 --> 00:48:09,000
u3.
And that is our constraint.

563
00:48:09,000 --> 00:48:12,000
My three variables,
u1, u2, u3, are constrained in

564
00:48:12,000 --> 00:48:14,000
this way.
The sum of this figure must be

565
00:48:14,000 --> 00:48:17,000
the area of a base.
And I want to minimize that guy.

566
00:48:17,000 --> 00:48:23,000
So that is my g and that guy
here is my f.

567
00:48:23,000 --> 00:48:28,000
Now we try to apply our
Lagrange multiplier equations.

568
00:48:28,000 --> 00:48:33,000
Well, partial f of a partial u1
is -- Well,

569
00:48:33,000 --> 00:48:36,000
if you do the calculation,
you will see it is one-half a1,

570
00:48:36,000 --> 00:48:43,000
u1 over square root of u1^2
plus h^2 equals lambda,

571
00:48:43,000 --> 00:48:46,000
what is partial g,
partial a1?

572
00:48:46,000 --> 00:48:50,000
That one you can do, I am sure.
It is one-half a1.

573
00:48:50,000 --> 00:49:00,000
Oh, these guys simplify.
If you do the same with the

574
00:49:00,000 --> 00:49:09,000
second one -- -- things simplify
again.

575
00:49:09,000 --> 00:49:17,000
And the same with the third one.
Well, you will get,

576
00:49:17,000 --> 00:49:21,000
after simplifying,
u3 over square root of u3

577
00:49:21,000 --> 00:49:24,000
squared plus h squared equals
lambda.

578
00:49:24,000 --> 00:49:27,000
Now, that means this guy equals
this guy equals this guy.

579
00:49:27,000 --> 00:49:33,000
They are all equal to lambda.
And, if you think about it,

580
00:49:33,000 --> 00:49:39,000
that means that u1 = u2 = u3.
See, it looked like scary

581
00:49:39,000 --> 00:49:42,000
equations but the solution is
very simple.

582
00:49:42,000 --> 00:49:45,000
What does it mean?
It means that our point q

583
00:49:45,000 --> 00:49:47,000
should be equidistant from all
three sides.

584
00:49:47,000 --> 00:49:52,000
That is called the incenter.
Q should be in the incenter.

585
00:49:52,000 --> 00:49:56,000
The next time you have to build
a golden pyramid and don't want

586
00:49:56,000 --> 00:49:59,000
to go broke, well,
you know where to put the top.

587
00:49:59,000 --> 00:50:03,000
If that was a bit fast, sorry.
Anyway, it is not completely

588
00:50:03,000 --> 00:50:06,000
crucial.
But go over it and you will see

589
00:50:06,000 --> 00:50:08,000
it works.
Have a nice weekend.