Second partial derivative test intuition Khan Academy English Korean

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- [Voiceover] Hey everyone.

안녕하세요

So in the last video I introduced

지난 영상에서는

this thing called the second partial derivative test,

이계 편미분 검사에 대해 배웠습니다

and if you have some kind of multivariable function

어떤 다변수 함수에 적용됩니다

or really just a two variable function

예를 들어 이변수 함수

is what this applies to, something that's

그러니까 x,y에 대해

f of x, y and it outputs a number.

함숫값을 출력하는 함수 말이죠

When you're looking for places where it has

함수의 그래프에서

a local maximum or a local minimum, the first step,

극대점이나 극소점을 찾을 때는

as I talked about a few videos ago,

이전 영상에서 말했듯이

is to find where the gradient equals zero

그래디언트가 0인 점

and sometimes you'll hear these called

critical points or stable points,

임계점을 찾으면 됩니다.

but inputs where the gradient equals zero

그리고 그때 변수의 값은

and that's really just a way of compactly

모든 편미분 값이

writing the fact that all the partial

0이라는 사실을

derivatives are equal to zero.

말해주는 것이죠

Now when you find a point like this,

이러한 점을 찾으면

in order to test whether it's a local maximum

이것이 극대 혹은 극소인지를

or a local minimum or a saddle point

그래프를 보지 않고

without actually looking at the graph,

판별할 수 있어야 합니다

'cause you don't always have the ability

항상 그래프를 볼 수 있는 것은 아니기 때문이죠

to do that at your disposal, the first step

그러기 위해 처음으로 할 일은

is to compute this long value,

이 긴 수식을 계산하는 것입니다

and this is the thing I wanna give intuition behind.

이 수식에 대한 직관을 말해드리고 싶군요

Where you take all three second partial derivatives,

3가지 이계 편미분을 구하면

the second partial derivative with respect to x,

즉 x에 관한 이계 편미분과

the second partial derivative with respect to y

y에 관한 이계 편미분,

and the mixed partial derivative

x에 대해 한 번,

where first you do it with respect to x,

y에 대해 한 번 미분한

then you do it with respect to y.

이계 편미분을 구하고

And you compute this value where you evaluate

앞서 구한 임계점에서

each one of those at your critical point

이 값을 구하고

and you multiply the two pure second partial derivatives

한 변수에 대해 두 번 미분한 것끼리 곱하여

and then subtract off the square

한 번씩 미분한 값의

of the mixed partial derivative

제곱을 빼 주면

and again, I'll give intuition for that in a reason,

이 이유에 대해서는

but right now we just kinda take it,

나중에 설명하기로 하고

oh, alright, I guess you compute this number

이 값을 구하여

and if that value H, if that value H

값을 H라 했을 때

is greater than zero, what it tells you,

H가 양수이면

what it tells you is that you definitely have

그 임계점이

either a maximum or a minimum.

극대 혹은 극소라는 말이 됩니다

So you definitely have either a maximum or a minimum.

반드시 극대, 극소를 가집니다

And then to determine which one

이 둘 중 무엇인지 알아보기 위해서는

you just have to look at the concavity in one direction.

한쪽에서의 오목, 볼록만 판별하면 됩니다

So you'll look at the second partial derivative

with respect to x for example,

x에 관한 이계 편미분을 보고

and if that was positive that would tell you

만약 이것이 양수이면

when you look in the x direction

x축 방향에서 보았을 때

there's a positive concavity, if it was negative

아래로 볼록이고, 음수이면

it would mean a negative concavity.

위로 볼록이라는 말입니다

And so that means a positive value

다시 말해 이계 편미분이

for that second partial derivative

would mean a local minimum and a negative value

극대를, 음수이면

would mean a local maximum.

극소를 나타낸다는 말입니다

So that's what it means if this value H

이것은 H값이

turns out to be greater than zero.

양수일 때의 이야기입니다

And if this value H turns out to be less than zero,

만약 H값이 음수이면

strictly less than zero, then you definitely

0과 같지는 않습니다

have a saddle point, saddle point.

이때는 안장점을 가집니다

Which is neither a maximum, nor a minimum.

극대나 극소는 아닙니다

It's kind of like there's disagreement

보는 방향에 따라

in different directions over whether

극대와 극소가

it should be a maximum or a minimum.

달라지는 점이죠

And if H equals zero, the test isn't good enough.

H값이 0이면 쓸모가 없습니다

You would have to do something else to figure it out.

판별하기 위해선 다른 식을 써야 하죠

So why does this work?

이것이 왜 될까요

Why does this seemingly random conglomeration

그냥 이계 편미분을 대충

of second partial derivatives give you a test

섞어놓은 듯한 식이

that let's you determine what type

이 정류점이 어떤 점인지

of stable point you're looking at?

알 수 있게 할까요

Well let's just understand each term individually.

각 항을 떼어서 생각해봅시다

So this second partial derivative with respect to x,

x에 대한 이계도함수를 보면

since you're taking both partial derivatives

두 번 다 x에 대해

with respect to x, you're basically treating

미분하는 것이기 때문에

the entire multivariable function as if x

이 다변수 함수 전체를

is the only variable and y was just some constant.

x에만 대한 함수로 생각하는 것입니다

So it's like you're only looking at

x축 방향으로의

movement in the x direction.

움직임만 보는 것이죠

So in terms of a graph, let's say we've got like,

그래프로 말하자면

this graph here, you can imagine slicing this

x축 방향의 움직임만 보기 위하여

with a plane that represents movement

그래프를 평면으로

purely in the x direction, so that'll be

자르는 것을 생각할 수 있습니다

a constant y value slice, and you take a look

y가 상수인 평면 말이죠

at the curve where this slice intersects your graph.

이 평면과 그래프가 이루는 교선을 봅시다

And in the one that I have pictured here

이 그래프에서 보면

it looks like it's a positive concavity.

아래로 볼록인 것 같습니다

So this term right here kind of tells you x concavity.

즉 이 항이 x 방향의 오목, 볼록을 나타내죠

So it's kind of like the, what is the concavity

다시 말해 x만을 변수로 했을 때

as far as the variable x is concerned.

오목, 볼록이 어떻게 되는가를 보게 됩니다

And then symmetrically, this over here,

마찬가지로

when you take the partial derivative

y에 대해서만 두 번

with respect to y two times in a row,

편미분을 하게 되면

it's like you're ignoring the fact

x가 변수라는 사실을

that x is even a variable and you're looking

고려하지 않고

purely at what movement in the y direction looks like.

y축 방향의 움직임만 보는 것입니다

Which on the graph that I have pictured here,

이 그래프에서는

also happens to give you kind of this

이쪽 교선도

positive concavity parabola look,

아래로 볼록한 모양이군요

but the point is that the curve on the graph

여기서 중요한 것은

that results from looking at movement

y축 방향의 움직임을

purely in the y direction can be analyzed

y에 대해 두 번

just looking at this partial derivative

편미분하기만 하면

with respect to y twice in a row.

알 수 있다는 것입니다

So that term kind of tells you y concavity, y concavity.

즉 이 항이 y의 오목, 볼록을 말해주는 것입니다

Now first of all, notice what happens

만약 이 둘이 다르다면

if these disagree.

어떻게 될까요

If say, x thought there should be positive concavity

예를 들어 x방향으로는 아래로 볼록인데

and y thought there should be negative concavity.

y방향으로는 위로 볼록인 상황이죠

Here, I'll write that down, what that means.

여기에 다시 적어보면

If x thinks there's positive concavity

x 방향으로 아래로 볼록이라면

we have here some kind of positive number

어떤 양수 값을 가지게 되고

that I'll just write as like, a plus sign in parenthesis.

(+)로 부호만 표시하겠습니다

And then this here, y concavity, would be

y축 방향의 오목, 볼록은

some kind of negative number, so we'll just put like,

음수 값을 가지게 되는데

a negative sign in parenthesis.

(-)의 부호를 가지겠군요

So that would mean this very first term

즉 이 첫 번째 항은

would be a positive times a negative

양수에 음수를 곱했으니

and that first term would be negative.

음수일 것입니다

And now the thing that we're subtracting off,

그리고 뒤에서 빼는 값을 보면

I'll get to the intuition behind this

이것이 무엇인지는

mixed partial derivative term in a moment,

나중에 설명하더라도

but for now you can notice that it's something squared,

무언가의 제곱

it's something that's always a positive term.

즉 양수임을 알 수 있습니다

So you're always subtracting off a positive term

즉 양수를 빼게 되는데요

which means if this initial one is negative,

첫 항이 음수이면

the entire term H is definitely gonna be negative,

H값은 당연히 음수일 것입니다

so it's gonna put you over into this saddle point territory.

그러니 안장점이 되는 것이죠

Which makes sense, because if the x direction

직관적으로도

and the y direction disagree on concavity

x와 y 방향의 오목, 볼록이 다르면

that should be a saddle point.

안장점이 생깁니다

The quintessential example here

is when you have the function f of x, y

x, y의 함수가

is equal to x squared minus y squared, x squared minus y squared.

x제곱에서 y제곱을 뺀 즉 f(x,y)=x²-y²일 때입니다

And the graph of that, by the way,

the graph of that would look like this

그래프를 그려 보면

where, let's see, so orienting myself here,

이쪽 방향에서 볼 때

moving in the x direction you have kind of,

x축 방향으로 움직이면

positive concavity which corresponds

아래로 볼록임을 알 수 있고

to the positive coefficient in front of x squared,

이는 x²의 계수가 양수임에서 알 수 있습니다

and in the y direction it looks like negative concavity.

y 방향에서 볼 때는 위로 볼록인데

Corresponding to that negative coefficient

y²의 계수가 음수인 것과

in front of the y squared.

동일합니다

So when there's disagreement among these,

다시 말해 이 둘이 다르면

the test ensures that we're gonna have a saddle point.

안장점이 생김을 알 수 있죠

Now what about if they agree, right,

만약 둘이 같다면

what if either it's the case that x thinks

예를 들어 x 방향에서도

there should be positive concavity

아래로 볼록이고

and y thinks there should be positive concavity,

y 방향에서도 아래로 볼록일 때

or they both agree that there should be,

아니면 둘 다 음수가 나올 때를

you know, negative concavity.

살펴보겠습니다

In either one of these cases,

두 경우 모두에 대해서

when you multiply them together they're positive.

이 둘을 곱하면 양수가 나옵니다

So it's kind of like saying, if you look

purely in the x direction or purely in the y direction,

x만 고려했을 때랑 y만 고려했을 때

they agree, that there should be, you know,

둘이 같다면 당연히

definitely positive concavity

아래로 볼록이거나

or definitely negative concavity.

위로 볼록이겠죠

So that entire first term is going to be positive.

즉 첫 항은 양수입니다

So it's kind of like a clever way

x와 y 방향으로 본

of capturing whether or not the x directions

결과가 같은지 알아보는 방법이

and y directions agree.

상당히 잘 먹히는군요

However, the reason that it's not enough

하지만 이것이 다가 아닌 이유는

is 'cause in either case we're still

subtracting off something that's always a positive term.

양수를 빼고 있기 때문입니다

So when you have this agreement

그러니 x와 y를 고려한 것이

between the x dicretion and the y direction

서로 잘 맞으면

it then turns into a battle between

this x, y agreement and whatever's going on

한 번씩 미분한 편미분값 사이의

with this mixed partial derivative term.

경쟁이 됩니다

And the stronger that mixed partial derivative term,

한 번씩 미분한 쪽이 커지면

the bigger this negative number,

더 큰 음수값을 지니므로

so the more it's pulling the entire value

H값이 음수가 되게끔

H towards being negative.

만들 것입니다

So let me see if I can give a little

왜 이런 현상이

bit of reasoning behind why this

한 번씩 미분한 편미분이

mixed partial derivative term is trying

안장점이 되게 하는지

to pull things towards being a saddle point.

이유를 설명하도록 하겠습니다

Let's take a look at the very simple function

f of x, y, is equal to x times y.

간단한 함수를 보겠습니다

So what that looks like graphically,

그래프로 이 식을

f of x, y equal x times y, is this.

표현하고 보면

It looks like a saddle point.

안장점 같습니다

So let's go ahead and look at it's partial derivatives.

여기에 편미분을 계산해 봅시다

So the first partial derivatives,

일계 편미분을 먼저 하자면,

partial with respect to x and partial

x와 y에 대한 편미분을 볼 때

with respect to y, well when you do it

x에 대한 편미분은

with respect to x, x looks like a variable,

x를 변수, y를 상수로 취급하여

y looks like a constant, it's just that constant y.

값은 y가 됩니다

And when you do it with respect to y

y에 대해서 편미분하면

it goes the other way around.

반대로 됩니다

Y looks like the variable, x looks like the constant

y가 변수, x가 상수이기 때문에

so the derivative is that constant x.

값은 x가 됩니다

Now when you take the second partial derivatives,

이계 편미분을 구할 때는

if you do it with respect to x twice in a row

x에 대해 연속 두 번 하면

you're differentiating this with respect to x,

이것을 x로 미분하기 때문에

that looks like a constant, so you get zero.

0이 나옵니다

And similarly, if you do it with respect to y

마찬가지로 y에 대해 두 번 하면

twice in a row, you're doing this

x를 y에 대해 미분하는데

and the derivative of x with respect to y,

x가 상수이므로

x looks like a constant, goes to zero.

값이 0이 나옵니다

But the important term, the one that we're

하지만 직관을 얻을

getting an intuition about here,

중요한 점은

this mixed partial derivative, first with respect to x

x와 y에 대해 미분한 편미분을

then with respect to y, well you can view it in two ways.

두 가지 방법으로 봐서

Either you take the derivative of this expression

이것을 y로 미분하여

with respect to y, in which case it's one,

1이 나오거나

or you think of taking the derivative of this expression

이쪽을 x로 미분하여

with respect to x, in which case it's also one.

마찬가지로 1이 나옵니다

So it's kind of like this function is a very pure

두 변수에 대해 각각 한 번씩 미분한

way to take a look at what this

값이 어떤 모양인지

mixed partial derivative term looks like.

알아볼 수 있는 예입니다

And the higher the coefficient here,

또한 계수가 높으면

if I had put a coefficient of, you know, three here

예를 들어 여기 계수가 3이라면

that would mean that the mixed partial derivative

한 번씩 미분한 값은

would ultimately end up being three.

3이라는 결과가 나옵니다

So notice, the reason that this looks like a saddle

안장처럼 보이는 이유는

isn't because the x and y directions disagree,

x와 y 방향이 달라서가 아닙니다

in fact if you take a look at pure movement

사실 x축 방향으로는

in the x direction it just looks like a constant.

상수처럼 움직입니다

The height of the graph along this plane,

이 평면을 따른 그래프의 높이

along this line here is just a constant

이 선을 따른 높이가 상수인 것은

which corresponds to the fact that the

x에 대해 두 번 편미분하였을 때

second partial derivative with respect to x

그 값이 0인 것과

is equal to zero.

대응됩니다

And then likewise, if you cut it with a plane

마찬가지로

representing a constant x value,

고정된 x좌표를 갖는 평면으로 자르면

meaning movement purely in the y direction,

y축에 관련된 움직임이 나오는데

the height of the graph doesn't really change along there,

여기서 그래프의 높이는 변하지 않습니다

it's constantly zero which corresponds

값이 항상 0인데, 이는

to the fact that this other partial derivative is zero.

다른 편미분 값이 0인 것과 일치합니다

The reason that the whole thing looks like a saddle

이것이 안장처럼 보이는 이유는

is 'cause when you cut it with a diagonal plane here,

비스듬하게 평면을 잘라보면

a diagonal plane, it looks like it has negative concavity.

위로 볼록해 보이지만

But if you were to chop it, you know, in another direction

다른 방향으로 잘라 보면

it would look like it has positive concavity.

아래로 볼록하기 때문입니다

So in fact, this xy term is kind of like a way

of capturing whether there's disagreement

대각선 방향으로 잘랐을 때

in the diagonal directions.

일치하는지를 보는 것입니다

And one thing that might be surprising at first

처음 들으면 놀랄 만한 사실이 있는데

is that you only need one of these

이계 편미분에 대한 정보가 하나여도

second partial derivatives in order

다른 대각선 방향의

to determine all of the information

정보를 파악하기에

about the diagonal directions.

충분하다는 것입니다

'Cause you can imagine, you know, maybe there's

왜나하면 두 가지 벡터를 따라

disagreement between movement along one certain vector

움직일 때가 서로 다르면

and movement along another and you would have

무수한 방향에 대해 일일이

to account for infinitely many directions

찾아야 하지 않나 하는

and look at all of them.

상상을 할 수도 있기 때문입니다

And yet evidently, it's the case that you only

또한 이것은

really need to take a look at this

한 번씩 미분한 것을 살펴보아야 하는

mixed partial derivative term.

유일한 경우입니다

You know, along with the original pure

second partial derivatives with respect

한 변수를 두 번씩 미분한

to x twice and with respect to y twice.

값과 함께 말이죠

But still, looking at only three different terms

하지만 세 값을 본다고

to take into account possible disagreement

수많은 방향에서의 불일치를

in infinitely many directions actually

해결할 수 있다는 것은

feels like quite the surprise.

여전히 놀라워 보입니다

And if you want the full, rigorous justification

이것이 어떻게 작용하는지에 대한

for why this is the case, why this

엄밀한 증명

second partial derivative test works

즉 왜 이 판정법이 성립하는지에 대한

and kind of, an airtight argument.

완전한 증명에 대해서는

I've put that in an article that you can find

관심있는 사람들을 위해

that kind of, goes into the dirty details

자세한 것까지 하나하나 정리한

for those who are interested.

것이 있습니다

But if you just want the intuition,

하지만 직관만을 원한다면

I think it's fine to think about

여기서 했던 것처럼

the fact that this mixed partial derivative

한 번씩 편미분한 값이

is telling you how much your function

함수 f(x,y)가

looks like the graph of f of x, y equal x times y.

f(x,y)=xy와 얼마나 닮았는지를 나타낸다고 생각해도 됩니다

Which is the graph that kind of captures

대각선 방향에 대한

all of the diagonal disagreement.

모든 불일치를 잡아낼 수 있게요

And then when you let that term,

그리고 한 번씩 편미분한 그 항을

that mixed partial derivative term,

x와 y에 대해

kind of compete with the agreement

각각 편미분한 항과

between the x and y directions,

경쟁시키면

you know, if they agree very strongly,

x와 y가 매우 잘 맞으면

you have to subtract off a very strong amount

이 값을 음수로 만들기 위해서는

in order to pull it back to being negative.

매우 큰 값을 빼야 합니다

So this battle back and forth, if it's pulled

즉 이 공방전에서 매우 큰 값을 빼면

to be very negative that will give you a saddle point,

안장점이 나타나고

if it doesn't pull hard enough, then the agreement

적당히 작은 값이 빠지면

between the x and y directions wins out

처음 항이 우세하여

and it's either a local maximum or a local minimum.

극대나 극소가 나타납니다

So hopefully that sheds a little bit of light

이 공식이 왜 성립하는지에 대한

on why this term makes sense and why it's a reasonable way

직관적인 이유가 될 수 있을 것입니다

to combine the three different second partial derivatives

또 왜 세 가지 세 가지 편미분 값을

available to you, and again, if you want the full

수식으로 만들면 성립하는지는

details, I've written that up in an article form.

다른 곳에 정리해 두었습니다

I'll see you next video.

다음 영상에서 뵙겠습니다

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