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DAVID JORDAN: Hello, and welcome
back to recitation.

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In this question, we're going
to be considering a contour

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plot, which is given
to us as followed.

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The values are not indicated.

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So the first thing that
we want to do

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is we want to identify--

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on this contour plot, there is
a unique saddle point, and we

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want to label that as point A,
and there are two points which

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are either a maximum
or a minimum.

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We can't actually tell because
the labels aren't on this

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contour plot, but we want to
go ahead and label those

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anyways: B and C. So they're
either maximum or minimum, but

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we can still find them and we
can still identify them.

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So that's the first part
of the problem.

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The second part is since this
doesn't have the values

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entered onto the graph, we want
to consider what possible

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configurations could we have?

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So the second question is: can
B and C both be maximal?

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And can we have B maximal
but C minimal?

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OK.

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And then in each of these
two cases, we want to

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sketch the 3D graph.

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So why don't you take some
time to work this out.

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Pause the video, and we'll check
back, and I'll show you

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how I solve this.

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Hello, and welcome back.

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So to get started, why don't we
answer the first question

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by writing the points right
on our original graph.

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So I'll just come over here.

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Now, when we're looking for a
minimum or a maximum on a

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contour plot, you know, the
thing that we should keep in

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mind is that a minimal or a
maximal always is going to be

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contained in concentric contours
that are either

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approaching the minimum
from below or--

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excuse me-- approaching the
maximum from below or the

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minimum from above.

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And so if we look here, we see
that these rings start to

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become concentric, and somewhere
in here, there's got

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to be either a maximum
or a minimum.

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Because, you know, inside this
little region here, the

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function doesn't pass through
another contour plot, so we

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have to find either a maximum
or a minimum inside the

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innermost ring.

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And similarly, we have to
find either a minimum

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or a maximum here.

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So let's just call this one
B and this one C. OK.

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Now, we also have a saddle point
A in this problem, and

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it's a little bit hard to
see in the contour plot.

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I think it'll be a little even
clearer when we draw a 3D

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graph, but basically what's
happening is the fact that you

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have these contours--

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so this contour is, after all,
the same as that contour.

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So the value of the function
here and here are the same,

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and yet, if we look at this
point and this point, the

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values, they'll either
go up or down.

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We don't know.

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Let's assume that they go up.

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So here, in this direction, the
values are going up, and

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in this direction, the values
are determined by this contour

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curve, and so somewhere in
this middle region here,

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there's got to be
a saddle point.

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I think this'll be even clearer

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when we draw our graph.

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So we have a saddle point
A in the middle there.

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And, in fact, this is really,
this is the general picture of

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what a saddle point is
going to look like.

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It's going to be when you have
two either maxima or minima

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rising out, and you have a
contour which is containing

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the point in the middle.

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So those are our points A, B and
C that we're going to be

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interested in.

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So now, the second question that
we have to consider is

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can B and C both be maximal?

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And the third question is can
B be maximal and C minimal?

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And actually, we'll answer both
of these questions by

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just sketching an example,
so that's how will we'll

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understand this.

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So why don't we see if we can
sketch an example where B and

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C are both maximal.

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So here's the start of my graph
in three dimensions, and

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if we want B and C to both be
maximal, then let me go ahead

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and draw the contour lines
that we have. So we have,

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first of all, we had this one,
and then we had another one,

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and then we had a peak, and
then we had a peak.

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So if we want to draw this in
three dimensions, then what we

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just need to do is we just need
to follow these contour

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plots up out of the plane
and into space.

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So this goes up, and then
there's a maximum, and then it

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comes back down along the
contour lines, and then it

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goes back up, and then
it goes back down.

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So that's just one of the curves
that lies on the graph

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of the function, but then we
need to flush out the contour

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lines, which they look like
these sort of rings here.

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OK?

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And so, indeed, we do see that
it's possible for both B and C

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to be a maximal.

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Here's an example
of such a thing.

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And so here's our point B,
here's our point C, and as

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promised, I think it's much
clearer now how A becomes a

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saddle point.

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Because you have these two
mountains rising up, and the

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valley in between them
is necessarily a

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little saddle here.

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It increases in this
direction and it

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decreases in this direction.

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Now, for the second one, we're
asked can B be maximal and yet

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C be minimal?

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And the answer is still yes.

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And a nice way to think about
this problem, to think about

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the graph that I'm going to
draw, is imagine that we start

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over here and we dig a hole, and
as we're digging, we throw

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the dirt over behind us.

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So we're going to have a hole,
a dip here, and then we're

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going to pile that hole
up over here.

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And so then--now, notice that
both of these that I've drawn,

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if we don't label the contour
lines, these have the same

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contour plot, because the
concentric rings on C, which

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are telling us that the function
in increasing,

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they're the same below, because
this is essentially

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the same thing.

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The concentric rings that on
this curve, on this surface,

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were telling us that B was a
maximal point, now the same

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rings are telling us that
B is a minimal point.

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So this illustrates that a
contour plot is really--

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doesn't tell you everything
about the graph unless you

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actually label the values
of the contours.

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So here we see two examples
where the sort of global

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behavior of B and C are very
different even though they

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have the same contour plot.

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Now notice, in both cases,
A is a saddle point.

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It's increasing in
one direction and

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decreasing in the other.

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And I think I'll leave
it at that.

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