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DAVID JORDAN: Hello, and
welcome back to recitation.

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In this question,
we're going to be

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considering a
contour plot, which

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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--
on this contour plot,

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there is a unique
saddle point, and we

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want to label that
as point A, and there

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are two points which are
either a maximum or a minimum.

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

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on this contour
plot, but we want

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to go ahead and label
those anyways: B and C.

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So they're either
maxima or minima,

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but 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 entered

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onto the graph, we
want to consider

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what possible 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,

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we want to 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,

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and I'll show you
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

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on a contour plot,
you know, the thing

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that we should keep in mind
is that a minimal or a maximal

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always is going to be contained
in concentric contours that

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are either approaching
the minimum from below

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or-- excuse me-- approaching
the maximum from below

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or the minimum from above.

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

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to become concentric,
and somewhere in here,

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there's got to be either
a maximum or a minimum.

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Because, you know, inside
this little region here,

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the function doesn't pass
through another contour plot,

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so we have to find either
a maximum or a minimum

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inside the innermost ring.

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And similarly, we have to find
either a minimum or a maximum

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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,

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and 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

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is the fact that you
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

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and this point, the 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,

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and in this
direction, the values

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are determined by
this contour curve,

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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 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

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is the general picture
of what a saddle point is

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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

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which is containing the
point in the middle.

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So those are our
points A, B and C

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that we're going to
be interested in.

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So now, the second question
that we have to consider

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is 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

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by just sketching an
example, so that's how will

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we'll understand this.

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

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B and C are both maximal.

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

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and if we want B and C
to both be maximal, then

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let me go ahead and draw the
contour lines that we have.

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So we have, first of
all, we had this one,

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and then we had another
one, and then we had a peak,

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and then we had a peak.

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So if we want to draw
this in three dimensions,

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then what we just
need to do is we just

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need to follow these contour
plots up out of the plane

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and into space.

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

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and then it comes back down
along the contour lines,

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and then it goes back up,
and then it goes back down.

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So that's just one
of the curves that

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lies on the graph
of the function,

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but then we need to flesh
out the contour lines, which,

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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,

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and as promised, I think
it's much clearer now

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how A becomes a saddle point.

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Because you have these
two mountains rising up,

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and the valley in between
them is necessarily

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a 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

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can B be maximal and
yet 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,

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to think about the graph
that I'm going to draw,

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is imagine that we start
over here and we dig a hole,

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and as we're digging, we
throw the dirt over behind us.

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

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and then we're going to
pile that hole up over here.

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And so then--

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Now, notice that both of
these that I've drawn,

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if we don't label
the contour lines,

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these have the
same contour plot,

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because the
concentric rings on C,

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which 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

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the same 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

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unless you actually label
the values of the contours.

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So here we see
two examples where

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the sort of global
behavior of B and C

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are very different even though
they have the same contour

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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 decreasing in the other.

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And I think I'll
leave it at that.