
English: 
- [Voiceover] So we're told
let g be a differentiable function
defined over the closed interval
from negative six to six.
The graph of its derivative,
so they're giving the
graphing the derivative of g,
g prime is given below.
So this isn't the graph of g.
This is the graph of g prime.
What is the x value of the
left-most inflection point,
inflection point in the graph of g.
So they want, they don't
want to know the x value
of the inflection points
in the graph of g prime,
in this graph.
They want to know the inflection points,
the x values of the inflection
points, in the graph of g.
And we have to figure
out the left-most one.
So, let me just make a little table here,
to think about what is
happening at inflection points
in our second derivative,
our first derivative,
and our actual function.
So, this is g prime, prime.
This is g prime.
And this is our actual, I
guess you could call it,
the original function.
So an inflection point are points
where our second derivative
is switching sides.

Thai: 
เขาบอกเราว่า
ให้ g เป็นฟังก์ชันที่หาอนุพันธ์ได้
นิยามตลอดช่วงปิดจากลบ 6 ถึง 6
กราฟของอนุพันธ์ของมัน
เขาให้กราฟของอนุพันธ์ของ g
g ไพรม์ให้มาข้างล่าง
นี่ไม่ใช่กราฟของ g
นี่คือกราฟของ g ไพรม์
ค่า x ของจุดเปลี่ยนเว้าซ้ายสุด
จุดเปลี่ยนเว้าในกราฟของ g คืออะไร
เขาอยาก เขาไม่ได้อยากรู้ค่า x
ของจุดเปลี่ยนเว้าในกราฟของ g ไพรม์
ในกราฟนี้
เขาอยากรู้จุดเปลี่ยนเว้า
ค่า x ของจุดเปลี่ยนเว้า ในกราฟของ g
และเราตอ้งหาจุดซ้ายสุด
ขอผมสร้างตารางเล็กๆ ตรงนี้
เวลาคิดว่าเกิดอะไรขึ้นที่จุดเปลี่ยนเว้า
ในอนุพันธ์อันดับสอง อนุพันธ์อันดับหนึ่ง
และฟังก์ชันจริงของเรา
นี่คือ g ไพรม์ ไพรม์
นี่คือ g ไพรม์
และนี่คือของจริง คุณจะเรียกมัน
ว่าฟังก์ชันดั้งเดิมก็ได้
จุดเปลี่ยนเว้าคือจุด
ที่อนุพันธ์อันดับสองเปลี่ยนข้าง

Bulgarian: 
Дадена ни е диференцируемата
функция g(х),
дефинирана в затворения
интервал от –6 до 6.
Графиката на нейната производна –
тук ни е дадена графиката на
производната на g отдолу,
това е g'.
Това не е графика на
функцията g.
Това е графиката на g'.
Каква е стойността на х 
в най-лявата инфлексна точка
на графиката на g.
Тоест искат да знаят стойността
на х
в инфлексните точки на
графиката на g'.
Искат да знаят инфлексните точки,
стойностите на х в
инфлексните точки на графиката на g.
Да определеми коя
е най-лявата от тях.
Тук ще направя една таблица,
за да видим какво се случва
в инфлексните точки.
Втора производна,
първа производна,
действителна функция.
Това е g''.
Това е g'.
А това е истинската, ако можем
да кажем така,
оригиналната функция.
Инфлексни точки са точките,
в които втората
производна си сменя знака.

Korean: 
-6에서 6까지의 폐구간에서 정의된
미분 가능한 함수 g를
생각해보겠습니다
도함수의 그래프는
g'의 그래프이며
아래와 같이 주어져 있습니다
이것은 g의 그래프가 아닙니다
g'의 그래프입니다
가장 왼쪽의 변곡점의 x 값은
g의 그래프에서 무엇일까요?
즉 그들은 g' 그래프에서의
변곡점의 x값을 원하는 것이
아닙니다
그들은 g의 그래프에서
변곡점의 x좌표를 알고 싶어합니다
우리는 가장 왼쪽의 것을 알아내야 합니다
작은 표를 생각해보면
변곡점에 대해서
이계 도함수와 일계 도함수,
그리고 원래 함수입니다
이것이 g''이며
이것이 g'입니다
그리고 이것이 원래 함수입니다
원함수라 명칭합니다
변곡점이란
이계 도함수가 변하는 부분이며

Thai: 
มันจะเปลี่ยนจากบวกเป็นลบ หรือลบเป็นบวก
ลองพิจารณากรณีแรกกัน
g ไปจากบวกเป็นลบ
บวก บวกเป็นลบ
ถ้า g ไพรม์ไพรม์
ถ้าอนุพันธ์อันดับสองไปจากบวกเป็นลบ
อนุพันธ์อันดับหนึ่งจะเป็นอย่างไร?
นึกดู อนุพันธ์อันดับสอง
คืออนุพันธ์ของอนุพันธ์อันดับหนึ่ง
เมื่ออนุพันธ์อันดับสองเป็นบวก
เมื่ออนุพันธ์อันดับสองเป็นบวก
นั่นหมายความว่าอนุพันธ์อันดับหนึ่งจะเพิ่มขึ้น
ถ้าอนุพันธ์อันดับสองเปลี่ยนจากบวกเป็นลบ
นั่นหมายความว่า อันุพันธ์อันดับหนึ่ง
จะเปลี่ยนจากเพิ่มเป็นลด
จากเพิ่มเป็นลด
แล้วตัวฟังก์ชันเอง
เมื่ออนุพันธ์อันดับสองเป็นบวก
เราจะได้ มันหมายความว่า
มันหมายความว่า ความชันจะเพิ่มขึ้นเรื่อยๆ
แล้วนั่นหมายความว่า เรากำลังเว้าขึ้น
เว้าขึ้น

Bulgarian: 
Тя преминава от положителна
в отрицателна или обратно.
Да приемем, че това
е първият сценарий.
Значи g става
от положителна отрицателна
Положителна към отрицателна.
Ако g'' (два пъти прим),
ако втората производна
става от положителна отрицателна,
какво се случва с
първата производна?
Спомни си, втората
производна
е производна на първата
производна.
Когато втората производна
е положителна,
там където втората производна
е положителна,
това означава, че първата
производна нараства.
Ако втората производна преминава
от положителна към отрицателна,
това означава, че първата
производна
преминава от нарастване
към намаляване.
И самата функция,
когато втората производна
е положителна,
това означава, че
наклонът е постоянно
нарастващ.

English: 
It's going from positive
negative or negative to positive.
So, let's consider that first scenario.
So g, so going from positive to negative.
Positive, positive, to negative.
So if g prime prime,
if the second derivative's
going from positive to negative,
what is the first derivative doing?
Well, remember, the second derivative,
is the derivative of the first derivative.
So, where the second
derivative is positive,
where the second derivative is positive,
that means that the first
derivative is increasing.
So, if second derivative's
going from positive to negative,
that means first derivative
is going from increasing to decreasing.
From increasing to decreasing.
And the function itself,
well when the second
derivative is positive,
we are going to be, that means,
that means that the slope
is constantly increasing.
And so that means that
we are concave upwards.
So, concave upwards.

Korean: 
양에서 음으로 혹은
 음에서 양으로 변할 수 있습니다
첫번째 경우에 대해 생각해보겠습니다
g는 양에서 음으로 변합니다
양에서 음으로
결국 만약 g''
즉 이계 도함수가 양에서 음으로 변한다면
일계 도함수는 어떻게 될까요?
이계 도함수가 일계 도함수의
도함수라는 것을 생각해보겠습니다
즉, 이계 도함수가
양인 곳에서는
일계 도함수가 증가합니다
따라서 만약 이계 도함수가 
양에서 음으로 변한다면
그것은 일계 도함수가
증가하다가 감소함을
의미합니다
그리고 함수 자체는
이계 도함수가 증가할때는
그것의 기울기가
계속하여 증가합니다
그리고 이는 위로 오목을
의미합니다

English: 
Upwards to downwards.
To concave, to concave downwards.
But they've given us the graph of g prime.
So let's focus on what are the points
where g prime is going from
increasing to decreasing.
So let's see.
G prime is increasing,
increasing, increasing,
increasing, increasing at a slower rate,
and then it starts decreasing.
So, right over there
it's going from increasing to decreasing.
So then it's decreasing,
decreasing, decreasing.
Then it goes increasing,
increasing, increasing,
increasing, and then decreasing again.
So that's another point where we're going
from increasing to decreasing.
And those are the only ones
that look like we're going
from increasing to decreasing.
But we're not done yet.
Because it's not just about going from
the second derivative going
from positive to negative,
it's also the other way around.
Any time the second
derivative is switching signs.
So, it's also the situation
where we're going from
negative to positive.

Korean: 
위에서 아래로,
위로 오목에서 아래로 오목으로
그러나 g'의 그래프를 주었기에
g'이 증가할지 감소할지에 대해
집중해보겠습니다
한번 보겠습니다
g'은 증가하고 있습니다
느린 속도로 증가하다가
감소하기 시작합니다
저 부분에서
증가하다가 감소합니다
그래서 이것이 감소하며
증가하다가
다시 감소합니다
따라서 저것은 증가하다가 감소하는
또다른 지점입니다
저것들은
우리가 증가하다가 감소하는 유일한 부분입니다
그러나 아직 끝나지 않았습니다
이것이 이계 도함수가 양에서 음으로
변하는 것에만 관련있는 것이 아니기에
다른 방식으로도 가능합니다
이계 도함수의 부호가 바뀌는 부분에서의
상황을 생각해보면
음에서 양으로 변하는 부분에서

Bulgarian: 
Това означава, че имаме
изпъкнала нагоре крива.
Нагоре към надолу.
Към изпъкнала надолу.
Но тук са ни дали графиката
на g'.
Да се фокусираме върху
точките, където
g' преминава от нарастване
към намаляване.
g' нараства, нараства,
нараства,
нараства с бавно темпо,
и после започва 
да намалява.
Значи ето тук точно
преминава от нарастване
към намаляване.
Намалява, намалява.
После нараства, нараства,
нараства и отново започва
да намалява.
Това е друга точка, в която
преминава от нарастване
към намаляване.
Това са единствените, които
където преминава от
нарастване към намаляване.
Но още не сме готови.
Защото не става въпрос
само за
преминаването на втората
производна от положителна към отрицателна,
възможно е и в обратния случай.
Винаги, когато втората
производна си сменя знака.
Това е случаят,
в който отиваме от отрицателно
към положително.

Thai: 
ขึ้นไปลง
เว้า เว้าลง
แต่เขาให้กราฟของ g ไพรม์มา
ลองสนใจว่าจุดใด
ที่ g ไพรม์จะเปลี่ยนจากเพิ่มเป็นลด
ลองดูกัน
g ไพรม์กำลังเพิ่มขึ้น เพิ่มขึ้น เพิ่มขึ้น
เพิ่มขึ้น เพิ่มขึ้นด้วยอัตราที่ช้าลง
แล้วมันก็เริ่มลดลง
ตรงนี้
มันจะเปลี่ยนจากเพิ่มเป็นลด
แล้วมันลดลง ลดลง ลดลง
แล้วมันก็เพิ่มขึ้น เพิ่มขึ้น เพิ่มขึ้น
เพิ่มขึ้น แล้วก็ลดลงอีก
นั่นคืออีกจุดที่เราจะ
ได้จากเพิ่มเป็นลด
และพวกมันคือจุด
ที่ดูเหมือนว่า เราจะเปลี่ยนจากเพิ่มเป็นลด
แต่เรายังไม่เสร็จ
เพราะมันไม่ใช่แค่เปลี่ยนจาก
อนุพันธ์อันดับสอง จากบวกเป็นลบ
มันยังกลับกันได้
ทุกครั้งที่อนุพันธ์อันดับสองเปลี่ยนค่า
มันยังมีกรณี
ที่เราเปลี่ยนจากลบเป็นบวก

Bulgarian: 
За първата производна
това е преминаване от
намаляване към нарастване.
Да видим, намалява, намалява,
намалява и после нараства.
Това е точно тук.
После нараства, намалява,
намалява и пак нараства.
Ето тук.
Това са инфлексните точки,
които определихме визуално.
Ако погледнем възможните
отговори,
за да отговорим на
първоначалния въпрос,
най-лявата стойност на х
е равно на –3?
Това е за х = –3.
х = –1 е стойност на х,
за която имаме инфлексна точка.
Да видим, х = 2 е друг вариант,
и също х = 4.
Те са дали всички тези
инфлексни точки.
Но искат да посочим
най-лявата.

Korean: 
혹은 일계 도함수가 감소하다가
증가하는
감소하다가 증가하는 부분입니다
감소하다가 증가하는 부분을
보겠습니다
바로 저기입니다
이제 증가하고 감소합니다
감소하다가 증가합니다
따라서 저 부분에서
변곡점임을
알 수 있습니다
여러분이 이 것을 본다면
원래 문제에 대답하고 싶어질 것입니다
가장 왼쪽의 x가 -3인가요?
x가 -3과 같습니다
x가 -1인 부분은
변곡점입니다
x가 2인 부분과
4인 부분도 마찬가지입니다
결국 이들 모두는 변곡점입니다
그리고 그들은 가장
왼쪽의 것을 원한 것입니다

English: 
Or, for the first derivative
is going from decreasing,
decreasing to increasing.
Decreasing to increasing.
Well let's see we're
decreasing, decreasing,
decreasing, and then
we're increasing, alright.
So it's right there.
And then we're increasing, decreasing,
decreasing, decreasing,
and then we're increasing.
So right over there.
So these are the inflection points
that I've just figured out visually.
So, if you look at the choices,
if we want to answer
the original question,
well the left-most one is it
x is equal to negative three?
It's x equals negative three.
X equals negative one is indeed a x value,
where we have an inflection point.
And let's see, x equals two is one,
and so is x equals four.
So they actually listed, all
of these are inflection points.
And they just wanted the left-most one.

Thai: 
หรือ อนุพันธ์อันดับหนึ่งเปลี่ยนจากลด
ลดเป็นเพิ่ม
ลดเป็นเพิ่ม
ลองดู เราจะลดลง ลดลง
ลดลง แล้วเราจะเพิ่มขึ้น เอาล่ะ
มันคือตรงนี้
แล้วเราเพิ่มขึ้น ลดลง
ลดลง ลดลง แล้วเราก็เพิ่มขึ้น
ตรงนี้
พวกนี้คือจุดเปลี่ยนเว้า
ที่ผมหาได้จากภาพ
ถ้าคุณดูตัวเลือก
ถ้าเราอยากตอบคำถามเดิม
ตัวซ้ายสุด คือ x เท่ากับลบ 3 ใช่ไหม?
มันคือ x เท่ากับลบ 3
x เท่ากับลบ 1 เป็นค่า x
ที่เรามีจุดเปลี่ยนเว้าจริง
แล้วลองดู x เท่ากับ 2 ก็ใช่
เช่นเดียวกับ x เท่ากับ 4
เขาใส่มา ทั้งหมดนี้เป็นจุดเปลี่ยนเว้าหมด
และเขาอยากได้จุดที่อยู่ซ้ายสุด
