- [Voiceover] In the last video
we took a look at this function
f of x, y equal x to the fourth
minus four x squared plus y squared
which has the graph that
you're looking at on the left.
And we looked for all of the points
where the gradient is equal to zero.
Which basically means
both partial derivatives
are equal to zero and we solved that
and we found that there
were three different points.
The origin zero, zero,
and then square root
of two, zero and negative
square root of two, zero
which corresponds to this origin here
which is a saddle point and
then these two local minima.
And it seemed like we had
a reasonable explanation
for why this is a saddle point
and why both of those are local minima.
'Cause we took the
second partial derivative
with respect to x, that
was kind of all over
the board here, second partial derivative
with respect to x, and found that when you
evaluate it at x equals zero
you get a negative number,
kind of indicating a negative concavity
so it should look like a maximum.
And then when you do the same with
the second partial derivative of y,
man I always do this, I
always leave out the squared
on that lower term there, okay,
so second partial
derivative with respect to y
you get two as a constant, a positive.
And that kind of indicates that it looks
like a minimum to y.
So that's why, you know, this origin point
looks like a saddle point
'cause the x direction
and y direction disagree
and when you do this
with the other points
they kind of both agree
that it should look like a minimum.
But I said that's not enough, I said
that you need to take into account
the mixed partial derivative term.
And to see why that's true let me go ahead
and pull up another example for you here.
So the graph of the function
that you're looking at
right now, it clearly has a saddle point
at the origin that we can see visually,
but when we get the
equation for this function,
the equation is f of x,
y is equal to x squared
plus y squared minus four times xy.
Now let's go ahead and analyze
the partial differential information
of this function, we'll just take
it's partial derivatives, so the partial
with respect to x is equal to,
so we've got, when we
differentiate this term
we get two x, two x, y
looks like a constant,
we do nothing, and then this last term
looks like negative four times y
'cause y looks like a
constant, so negative four y.
And when we do the partial derivative
with respect to y, very
similarly we're gonna get two and
when we differentiate y squared.
Two y, and now we subtract minus four x
'cause x looks like the constant and y
looks like the variable, minus four x.
Now when we plug in x and
y are each equal to zero,
you know, we plug in the origin point
to both of these functions, we see
that they're equal to zero.
'Cause x is zero, y is
zero, this guy goes to zero.
Similarly over here, that goes to zero.
So we will indeed get a flat
tangent plane at the origin.
But now let's take a look at
the second partial derivatives.
If we do the second
partial derivatives purely
in terms of x and y, so if we take
the pure second partial derivative of f
with respect to x squared what we get,
we look at this expression,
we differentiate
it with respect to x and we
get a constant positive two
because that y does nothing for us.
And then similarly, when
we take the second partial
derivative with respect to y,
always forget that squared on
the bottom, we also get
a constant positive two
because this x does nothing for us
when we take the derivative
with respect to y.
So we get constant positive two.
So this would suggest that, you know,
there's positive concavity
in the x direction,
there's positive concavity
in the y direction,
so it would suggest that, you know,
it looks like an upward smiley face
from all directions and it
should be a local minimum.
But when we look at the
graph this isn't true,
it's not a local minimum,
it's a saddle point.
So what this tells us is that these two
second partial derivatives aren't enough,
we need more information.
And what it kinda comes down to is that
this last term here, minus four xy,
oh, actually, I think I made a mistake,
I think I meant to make this plus four xy.
So let's see, plus four xy
which would influence these.
It actually won't make a difference,
it still gives a saddle point,
but anyway, we've got
this plus four xy term
that evidently makes a difference.
That evidently kind of, influences
whether this is a local
minimum or a maximum.
And just to give a loose intuition
for what's going on here.
If instead of writing four here, I wrote,
I'm gonna write the variable P, okay,
and P is just gonna be some number
and I'm gonna move that variable around.
I'm gonna basically let it
range from zero up to four.
So right now, as you're looking
at it it's sitting at four.
So I'm gonna pull it
back and kind of let it
range back to zero just to see
how this influences the graph.
And we see that once we
pull it back to zero,
we get something where it kinda reflects
what you would expect,
where from the x direction
it's a positive smiley
face, from the y direction
it's also a positive smiley
face and everything's good,
it looks like the local minimum and it is.
And even as you let P range more and more,
and here P is around,
I'm guessing right now
it's around 1.5, you get something
that's still a local minimum.
It's a positive concavity
in all directions,
but there's a critical point here
where as you're moving P, at some point
it kind of passes over and
turns it into a saddle point.
And again, this is
entirely the coefficient
in front of the xy term,
it has nothing to do
with the x squared or the y squared.
So at some point it kinda passes over
and from that point on everything
is going to be a saddle point.
And in a moment it'll become clear
that that critical point happens
when P is equal to two.
So right here it's gonna
be when P equals two,
it kinda passes from making
things a local minimum
to a saddle point.
And let me show you the
test which will tell us
why this is true.
So the full reasoning behind this test
is something that I'll
get to in later videos,
but right now I just want to,
kind of have it on the table
and teach you what it
is and how to use it.
So this is called the second
partial derivative test.
Second partial derivative
test, I'll just write
deriv test since I'm a slow writer.
And basically what it says is if you found
a point where the
gradient of your function
at this point, and I'll write it kind of
x not, y not is our point, if
you found where it equals zero
then calculate the following value.
You'll take the second partial derivative
with respect to x twice,
so here I'm just using
that subscript notation
which is completely
the same as saying, you know,
second partial derivative
with respect to x twice, just different
choice in notation, and you evaluate it
at this point, x not, y
not, then you multiply that
by the second partial
derivative with respect to y
evaluated at that same point, y not,
and then you subtract off
the mixed partial derivative,
the one where first you
do it with respect to x
then with respect to y,
or in the other order,
it doesn't really matter as long as you
take it with respect
to both the variables.
You subtract off that guy squared.
So what you do is you
compute this entire value
and it's gonna give you
some kind of number,
and let's give it a name, let's name it H,
and if it's the case that
H is greater than zero
then you have either a max or a min,
you're not sure which one yet
and it's a max or a min.
And you can tell whether it's a maximum
or a minimum basically by
looking at one of these
partial derivative with respect to x twice
or with respect to y twice and kind of
getting a feel for the
concavity there, right.
If this was positive it would indicate
kind of a smiley face concavity
and it would be a local minimum.
And the fact that this entire value H
is greater than zero is what you need
to tell you that you can just do that,
you can look at the concavity with respect
to one of those guys and that'll tell you
the information you need
about the entire graph.
But if H is less than zero,
if H is less than zero,
then you definitely have a saddle point.
Saddle point.
And if H is purely equal to zero,
if you get that unlucky
case, then you don't know,
then the second partial derivative test
isn't enough to determine.
But almost all cases
you'll find that either
it's purely greater than zero
or purely less than zero.
So as an example let's
see what that looks like
in the case of the specific
function we started with.
Where P is some constant
that I was letting
kind of, range from zero to four
when I was animating it
here, but you should just
think of P as being some number.
Well in that case, this value H
that we plug in, and let's
say we're plugging it in
at the origin, right, we're
analyzing at the origin.
Well we've already calculated
the partial derivative
with respect to x twice in
a row and y twice in a row
and both of those, when we computed those
were just constants two, they
were equal to two everywhere
and in particular they're
equal to two at the origin.
So we can go ahead and just plug in those
and we see that this is two times two
and then now we need to subtract off
the mixed partial derivative squared.
So if we go ahead and compute that
where we take the derivative
with respect to x and then and
or y and then x, let's
say we started with the
partial derivative with respect to x.
When we take this
derivative with respect to y
we're gonna get this constant term
that's sitting in front of the y.
But really it's whatever this
constant P happened to equal.
And you might be able
to see that just looking
at this function, that when you take
the mixed partial derivative it's gonna be
the coefficient in front of the xy term,
'cause it's kind of like, first you do
the derivative with respect
to x and the x goes away
and then with respect to
y and that y goes away
and you're just left with a constant.
So what you end up getting here
in the second partial derivative test,
when we take that value which is P,
which might equal four or zero or whatever
we happen to have it
as and we square that,
we square that, that's gonna be the value
that we analyze.
So in the case where P
was equal to zero, right,
if we go over here and we scale so that P
is completely equal to zero,
then our entire value H,
H would equal four, and
because H is positive
it's definitely a maximum or a minimum
and then by analyzing one of those
second partial derivatives
with respect to x or y
and seeing that it's positive concavity,
we would see, oh, it's
definitely a local minimum
'cause positive concavity
gives local minimum.
But in the other case where, let's say
we let P range such that P
is all the way equal to four,
in formulas what that means for us
when we let P equal four is we're taking
two times two, minus four squared.
We're taking two times two, minus 16.
What that would imply,
sorry about getting kinda
scrunched on the board
here, is that H is equal to,
let's see four minus 16, negative 12.
I'm just gonna erase this to
kind of, clear up some room.
So when P equals four,
this is negative 12.
And in fact, this kind of
explains the crossover point
for when it goes from being local minimum
to a saddle point.
It's gonna be at that point where
this entire expression is equal to zero.
And you can see that
happens when P equals two.
So over here the crossover point
when it kind of goes from being
a local minimum to a saddle
point is at P equals two.
And when P perfectly
equals two, let's see,
so about here, the second
partial derivative test
isn't gonna be enough to tell us anything.
It can't tell us it's definitely a max
and it can't tell us that
it's definitely a saddle point
and in this particular
case that that corresponds
to the fact that the
graph is perfectly flat
in one direction and a
minimum in another direction.
In other cases it might
mean something different
and I'll probably make a video
just about that special case when
this whole value is equal to zero.
But for now all that I want to emphasize
is what this test is
where you take all three
second partial derivatives and you kind of
multiply together the two pure
second partial derivatives where you do
x and then x, and then one
where you do y and then y,
you multiply those together and then
you subtract off that mixed
partial derivative squared.
And in the next video I'll try to give
a little bit more intuition for,
you know, where this
whole formula comes from,
why it's not completely random,
why taking this and analyzing
whether it's greater than zero
or less than zero is a
reasonable thing to do
for analyzing whether a
point that you're looking at
is a local minimum or a local
maximum or a saddle point.
See you then.
