
Bulgarian: 
Тук ни е дадена в жълто
 графиката на у равно на f от х.
След това тук в този бледоморав цвят
е дадена графиката на производната на f,
т.е. f' от х.
А след това ето тук, в синьо, 
е дадена графиката
на у е равно на втората производна 
на функцията f, т.е. f'' от х.
Следователно това е производната 
на тази функция,
на първата производна ето тук.
Вече разгледахме примери как може
да откриваме точки на минимум и максимум.
Очевидно, ако имаме графиката
 пред себе си,
не е трудно за човек да открие,
че това е точка на локален максимум.
Функцията може да приема 
по-високи стойности след това.
А също така и да открие, че това е 
точка на локален минимум.
Функцията може да приема 
по-ниски стойности след това.
Но видяхме, че дори и да не разполагаме 
с графиката пред себе си,
ако можем да намерим 
производната на функцията,
то може – или дори ако не можем
да намерим производната на функцията –
да открием тези точки 
на минимум или максимум.
А начинът, по който го направихме,
е като зададем въпроса: Кои са 
критичните точки за дадената функция?
Критичните точки са места, където 
производната на функцията

Korean: 
 
여기 노란색으로 그린 것은 y=f(x)의 그래프 입니다
여기 연보라색으로 그린 것은
y=f'(x)의 그래프입니다
 
여기 파란색으로 그린 것은
y=f''(x)의 그래프 입니다
즉 여기 그린 것이
이계도함수입니다
그리고 우리는 이미 예시를 봤습니다
최소와 최대점을 구하는 방법에 관한
사실 우리 앞에 그래프가 있으면
알아내는 것이 어렵지 않습니다
이 점은 최대점이고
함수는 나중에 더 큰 값을 가질 수 있습니다
이것은 극소점입니다
나중에 함수는 더 작은 값을 가질 수 있습니다
하지만 그래프 없이도
함수의 미분을 할 수 있다면
혹은 함수의 미분을
하지 못하더라도
이 점들이 최소와 최대점인 것을 알 수 있습니다
우리가 한 방법으로
이 함수의 임계점은 어디인가요?
임계점은 함수의 미분이

Thai: 
 
สิ่งที่เรามีตรงนี้สีเหลืองคือ
กราฟของ y เท่ากับ f ของ x
แล้วตรงนี้สีชมพู ผมได้
วาดกราฟ y เท่ากับอนุพันธ์ของ f
คือ f ไพรม์ของ x
แล้วตรงนี้สีฟ้า ผมได้วาดกราฟ
y เท่ากับอนุพันธ์อันดับสองของฟังก์ชันเรา
นี่คืออนุพันธ์ของตัวนี้
ของอนุพันธ์อันดับหนึ่งตรงนี้
และเราเห็นตัวอย่างไปแล้วว่าเรา
หาจุดสูงสุดและต่ำสุดได้อย่างไร
แน่นอน ถ้าเรามีกราฟตรงหน้า
มันไม่ยากนักที่สมองมนุษย์จะระบุ
ว่าอันนี้เป็นจุดสูงสุดท้องถิ่น
ฟังก์ชันอาจมีค่ามากขึ้นต่อไป
แล้วก็ระบุว่าจุดนี้คือจุดต่ำสุดท้องถิ่น
ฟังก์ชันอาจมีค่าต่ำลงต่อไป
แต่เราเห็นว่า ถึงแม้ว่าเราจะไม่มีกราฟนี้ตรงหน้า
ถ้าเราหาอนุพันธ์ของฟังก์ชัน
เราอาจ -- ถึงแม้ว่าเราจะไม่สามารถหา
อนุพันธ์ของฟังก์ชันได้ -- เรา
ก็อาจระบุจุดเหล่านี้เป็นจุดต่ำสุดหรือจุดสูงสุดได้
วิธีที่เราทำ คือเราบอกว่า โอเค
จุดวิกฤตของฟังก์ชันนี้คืออะไร?
จุดวิกฤตคือจุดที่อนุพันธ์ของฟังก์ชัน

Czech: 
Máme tady žlutě nakreslený
graf funkce y rovná se f(x).
Dále tu je světle fialový graf funkce
y rovná se derivace f,
tedy y rovná se
f(x) s čárkou,
a poté je tu v modré barvě graf funkce
y rovná se druhá derivace f.
Toto je tedy derivace
téhle první derivace.
Už jsme viděli, jak lze najít lokální 
minima a maxima funkce.
Samozřejmě, pokud před
sebou máme graf,
tak lehce poznáme, že zde má
funkce lokální maximum.
Funkce pak dále může
nabývat ještě vyšších hodnot.
Rovněž snadno nahlédneme,
že zde má funkce lokální minimum.
Funkce pak tady může
nabývat ještě nižších hodnot.
Viděli jsme ale, že i když
před sebou nemáme graf,
ale máme předpis funkce,
který jsme schopni zderivovat,
tak můžeme...
Možná i když funkci
nedokážeme zderivovat.
...tak můžeme zjistit, že tohle jsou body
lokálního minima nebo maxima.
Dělali jsme to tak, že jsme nejprve
našli stacionární body funkce.

Portuguese: 
O que tenho aqui em amarelo
é o gráfico de y igual a f de x.
E aqui em lilás fiz o gráfico de y igual
a derivada de f, que é f linha de x.
E aqui em azul, fiz o gráfico de y igual
a segunda derivada de nossa função.
Então essa é a derivada disso,
da primeira derivada que está aqui.
E já vimos exemplos de como podemos
identificar pontos mínimos e máximos.
Claramente, tendo o gráfico bem aqui
não é difícil para um cérebro humano
identificar que isso é um
ponto máximo local.
A função pode ter valores maiores
mais à frente.
E identificar isso como um
ponto mínimo local.
A função pode ter valores menores
mais à frente.
Mas vimos, mesmo não tendo
o gráfico na nossa frente
se formos capazes de ter a derivada
da função, talvez--
ou mesmo se não conseguimos
ter a derivada da função--
talvez possamos identificar esses
pontos como mínimo ou máximo.
Fizemos assim e, OK, quais são os
pontos críticos dessa função?
Bem, pontos críticos são onde
a derivada da função

English: 
What I have here in yellow is
the graph of y equals f of x.
Then here in this
mauve color I've
graphed y is equal to
the derivative of f
is f prime of x.
And then here in
blue, I've graphed
y is equal to the second
derivative of our function.
So this is the
derivative of this,
of the first derivative
right over there.
And we've already seen
examples of how can we
identify minimum
and maximum points.
Obviously if we have
the graph in front of us
it's not hard for a
human brain to identify
this as a local maximum point.
The function might take
on higher values later on.
And to identify this as
a local minimum point.
The function might take on
the lower values later on.
But we saw, even if we don't
have the graph in front of us,
if we were able to take the
derivative of the function
we might-- or even if
we're not able to take
the derivative of
the function-- we
might be able to identify these
points as minimum or maximum.
The way that we did
it, we said, OK,
what are the critical
points for this function?
Well, critical points are
where the function's derivative

Chinese: 
What I have here in yellow is the graph of y=f（x）.
That here in this move color I’ve graphed y’s equal
to the derivative of f, is f′(x).And then here
in blue I graphed y is equal to the second derivative of our function.
So this is the derivative of this, of the first
derivative right over there. And we’ve already seen examples of how
can we identify minimum and maximum points. Obviously, if we have a graph in front of us,
it’s not hard for human brain to identify this as a local
maximum point. The function might take on higher values later on. And
to identify this as a local minimum point. The function might take on
lower values later on. But we saw, even if we don’t have a graph in front of
us, if we are able to take the derivative of the function, we might…
or if we are not able to take the derivative of the function. We might be able to identify
these points as maximum or minimum. The way that we did it. Ok… what are the critical
points for this function. Well, critical points over the function where the function’s

Chinese: 
derivative is either undefined or zero. This the function’s derivative.
It’s zero here and here. So we would call those critical points.
I don’t see any undefined. Any point was the derivative’s undefined
just yet. So we would call here and
here, critical points. So these are candidate
minimum…these are candidate points which are function might take on a minimum or
maximum value. And the way that we figured out whether it was a minimum or maximum
value is to look at the behavior of the derivative around that point
and over here we saw the derivative is de...or the
derivative is positive.The derivative is positive
as we approach that point
and then it becomes negative. It goes from being positive
to negative as we cross that point which means that the function]
was increasing. If the derivative is positive that means the function was increasing
as we approach that point and then decreasing as we leave that point.
Which is a pretty good way to think about this… Being a maximum point,

Czech: 
To jsou body, ve kterých derivace funkce
buď není definovaná, nebo se rovná nule.
Tohle je derivace
naší funkce.
Nule se rovná tady a tady,
takže to budou stacionární body.
Nevidím žádný bod,
ve kterém by derivace nebyla definovaná.
Tyto dva body jsou
tedy stacionární body.
Jde o podezřelé body, ve kterých má funkce
nejspíš lokální minimum nebo maximum.
Zda jde o lokální minimum
nebo maximum jsme zjistili tak,
že jsme se podívali na chování
derivace na okolí daného bodu.
V tomto případě vidíme, že derivace
je kladná, když se k bodu blížíme zleva,
a pak se
stává zápornou.
Derivace se tedy změnila z kladné na
zápornou při průchodu tímto bodem,
což znamená,
že funkce byla rostoucí...
Derivace je kladná, takže funkce roste,
jak se k bodu blížíme zleva,
a pak funkce klesá,
jak jdeme napravo od tohoto bodu.
To znamená, že jde o bod
lokálního maxima.

Bulgarian: 
или не е дефинирана, или е равна на 0.
Това е производната на функцията.
Тук и ето тук е равна на 0.
Ще наречем тези места 
критични точки.
Засега не виждам точки, в които
производната не е дефинирана.
Тази и тази точка ще ги означим 
като критични точки.
Това са точки, които са кандидати 
за функцията,
за минимална или максимална
 стойност.
Начинът, по който определяме дали
е минимална или максимална стойност,
е като проверим поведението на производната 
около всяка една от тези точки.
Ето тук виждаме, че производната
 е положителна,
когато достигаме до тази точка.
А след това става отрицателна.
Променя се от положителна 
на отрицателна,
когато преминава през тази точка.
Което означава, че функцията 
е била нарастваща.
Ако производната е положителна,
това означава, че функцията е била
 нарастваща, когато се приближава
към тази точка, а след това е намаляваща, 
когато продължава след тази точка,
което е достатъчно добро 
основание да смятаме,
това е точка, в която има максимум.

Thai: 
ไม่นิยามหรือเป็น 0
นี่คืออนุพันธ์ของฟังก์ชัน
มันคือ 0 ตรงนี้และตรงนี้
เราเรียกพวกมันว่าจุดวิกฤตได้
และผมไม่เห็นจุดอื่นใดที่
อนุพันธ์ไม่นิยาม
เราจึงเรียกจุดนี้กับจุดนี้ว่าจุดวิกฤต
พวกมันเป็นจุดตัวแทนที่ฟังก์ชันของเรา
อาจมีค่าต่ำสุดหรือสูงสุด
และวิธีที่เราหาว่ามันคือ
จุดต่ำสุดหรือสูงสุดหรือไม่ คือ
ดูพฤติกรรมของอนุพันธ์รอบจุดนั้น
และตรงนี้เราเห็นว่าอนุพันธ์เป็นบวก
เมื่อเราเข้าหาจุดนั้น
 
แล้วมันกลายเป็นลบ
มันไปจากบวกเป็นลบ
เมื่อเราข้ามจุดนั้น
ซึ่งหมายความว่าฟังก์ชันกำลังเพิ่ม
ถ้าอนุพันธ์เป็นบวก
นั่นหมายความว่าฟังก์ชันจะเพิ่มขึ้นเมื่อเราเข้าหา
จุดนั้น แล้วลดลงเมื่อเราออกจากจุดนั้น
ซึ่งเป็นวิธีคิดที่ดีว่าจุดนี้
เป็นจุดสูงสุด

English: 
is either undefined or 0.
This is the
function's derivative.
It is 0 here and here.
So we would call
those critical points.
And I don't see
any points at which
the derivative is
undefined just yet.
So we would call here
and here critical points.
So these are candidate points
at which our function might
take on a minimum
or a maximum value.
And the way that we
figured out whether it
was a minimum or
a maximum value is
to look at the behavior of the
derivative around that point.
And over here we saw the
derivative is positive
as we approach that point.
And then it becomes negative.
It goes from being
positive to negative
as we cross that point.
Which means that the
function was increasing.
If the derivative
is positive, that
means that the function was
increasing as we approached
that point, and then decreasing
as we leave that point, which
is a pretty good way
to think about this
being a maximum point.

Korean: 
정의되지 않거나 0인 점입니다
이것이 함수의 미분입니다
여기랑 여기서 0입니다
우리는 이 점들을 임계점이라고 할 수 있습니다
저는 아직 미분값이 정의되지 않은 점을
찾지 못했습니다
그래서 우리는 이 점들을 임계점이라고 할 수 있습니다
그래서 이 점들이 후보점들입니다
함수값이 최대 혹은 최소가 될 수 있는
우리가 저 점에서
최소 혹은 최댓값을 갖는지 아는 방법은
그 점 주변에서의 미분값을 살펴보는 것입니다
여기서 미분값이 양수인 것을 알 수 있습니다
이 점에 접근하면서
 
그리곤 음수가 됩니다
미분값이 양수에서 음수가 됩니다
이 점을 지나면서
미분값이 양수라는 것은
함수가 양수라는 것을 의미하고
그것은 함수가 증가한다는 것을 의미합니다
저 점에 도달하면서 그리고 저 점을 떠나면서 음수가 됩니다
이것은 꽤 좋은 방법입니다
최대점이 되는

Portuguese: 
é ou indefinida ou zero.
Essa é a derivada da função.
É zero aqui e aqui.
Então chamamos de pontos críticos.
E não vejo pontos onde a derivada
é indefinida por enquanto.
Então chamamos aqui e aqui
de pontos críticos.
Esses são pontos candidatos
onde a função pode ter
valor mínimo ou máximo.
E da forma que descobrimos quando
seria valor mínimo ou máximo foi
olhando o comportamento da derivada
em torno desse ponto.
E por aqui vemos que a derivada é positiva
quando nos aproximamos desse ponto.
E então se torna negativa.
Começa sendo positiva e fica negativa
quando cruzamos esse ponto.
O que significa que a função
estava crescendo.
Se a derivada é positiva, significa que
a função estava crescendo quando
nos aproximamos desse ponto,
e decrescendo quando
deixamos esse ponto,
que é uma boa maneira de pensar
nisso sendo um ponto máximo.

Thai: 
ถ้าเราเพิ่มขึ้นเมื่อเราเข้าใกล้มัน
และลดลงเมื่อเราออกจากมัน แล้วจุดนั้น
จะเป็นจุดสูงสุดแน่นอน
เช่นเดียวกัน ตรงนี้เราเห็น
ว่าอนุพันธ์เป็นลบเมื่อเราเข้าใกล้จุดนั้น
ซึ่งหมายความว่าฟังก์ชันกำลังลดลง
และเราเห็นว่าอนุพันธ์เป็นบวก
เมื่อเราออกจุดนั้น
เราไปจากอนุพันธ์เป็นลบ
ไปยังอนุพันธ์เป็นลบ ซึ่งหมายความว่า
ฟังก์ชันจะไปจากลดเป็นเพิ่ม
รอบจุดนั้น ซึ่งเป็นสัญญาณที่ดี
หรือมันเป็นการบ่งชี้ ว่าจุดวิกฤตนี้
เป็นจุดที่ฟังก์ชันมีค่าต่ำสุด
สิ่งที่ผมอยากทำตอนนี้ คือขยายสิ่งต่างๆ
โดยใช้แนวคิดเรื่องความเว้า หรือคอนเควิตี้
 
และผมรู้ว่าผมออกเสียงผิด
มันอาจเป็น คอนแควิตี้
แต่เวลาคิดถึงความเว้า เรา
จะเริ่มดูอนุพันธ์อันดับสอง แทนที่จะ
มองเป็นแค่การเปลี่ยนผ่าน เวลาคิดว่าจุดนี้

English: 
If we're increasing
as we approach it
and decreasing as we
leave it, then this
is definitely going
to be a maximum point.
Similarly, right
over here we see
that the derivative is negative
as we approach the point, which
means that the
function is decreasing.
And we see that the
derivative is positive
as we exit that point.
We go from having a
negative derivative
to a positive
derivative, which means
the function goes from
decreasing to increasing
right around that point, which
is a pretty good indication,
or that is the indication,
that this critical point is
a point at which the function
takes on a minimum value.
What I want to do
now is extend things
by using the idea of concavity.
And I know I'm
mispronouncing it.
Maybe it's concavity.
But thinking about
concavity, we could
start to look at the second
derivative rather than kind
of seeing just this transition
to think about whether this

Czech: 
Když se k bodu blížíme zleva, funkce
roste, a když jdeme doprava, tak klesá.
Tohle je tedy určitě
lokální maximum.
Podobně, když se podíváme sem,
tak vidíme, že derivace je záporná,
jak se k tomu bodu blížíme zleva,
což znamená, že funkce klesá.
Vidíme, že když jdeme od tohoto
bodu doprava, tak je derivace kladná.
Derivace se změnila
ze záporné na kladnou,
což znamená, že funkce se v tomto bodě
změnila z klesající na rostoucí,
což nám říká, že v tomto stacionárním
bodě má funkce lokální minimum.
Rád bych teď naše znalosti
rozšířil o konvexitu funkce.
Vím, že to možná
špatně vyslovuji.
Abychom pochopili konvexitu funkce,
podívejme se na druhou derivaci a na to,
jak díky ní namísto těchto změn
při průchodu bodem poznáme,

Bulgarian: 
Ако функцията е нарастваща, 
когато се приближаваме към нея,
а намаляваща след това, то тогава
това определено е точка, 
в която има максимум.
Подобно на това, 
точно ето тук виждаме,
че производната е отрицателна, когато
 се приближаваме към тази точка,
което означава, че функцията 
е намаляваща.
И виждаме, че производната 
е положителна,
когато продължава след тази точка.
Производната се променя 
от отрицателна
към положителна, което означава,
че функцията се променя 
от намаляваща към нарастваща,
около тази точка. Това е достатъчно
 добър показател,
или това е показателят, че 
тази критична точка
е точка, в която функцията достига 
до минимална стойност.
Това, което искам да направя сега, 
е да разширя тази тема,
като използвам идеята за вдлъбнатост.
Вдлъбнатост.
Знам, че не го произнасям правилно.
Може би е вдлъбнатост.
Като мислим за вдлъбнатост,
може да разгледаме 
втората производна не като
просто пресичане на оста х, а дали това

Chinese: 
for increasing as we approach and decreasing as we leave it. Then this is definitely going
to be a maximum point. Similarly,
right over here, we see that the function is negative or the derivative
is negative as we approach the point which means that the
function is decreasing. And we see the derivative is
positive as we exit that point. We go for having a negative derivative to a positive
derivative which means the function goes from decreasing to
increasing right around that point, which is a pretty good indication.
Or that is an indication, that this critical point is a point at which the
function takes on a minimum…a minimum value.
What I want do now is to extend things by using
the ideal of concavity… con-ca[ei]-vity.
And I know I’m mispronouncing it, maybe it’s conca[æ]vity,
but new thinking about concavity. Start to look at the second
derivative, it rather than kind of seeing just as transition. To think about

Portuguese: 
Se crescemos quando nos aproximamos
e decrescemos quando no afastamos,
então isso será um ponto máximo.
Similarmente, bem aqui podemos ver
que a derivada é negativa quando
nos aproximamos do ponto,
que significa que a função
está decrescendo.
E vemos que a derivada é positiva
quando saímos desse ponto.
Partimos de uma derivada negativa
para uma positiva, que significa
que a função começa decrescendo
e cresce bem em torno desse ponto,
o que é uma boa indicação,
ou que é a indicação,
que esse ponto crítico é onde
a função tem um valor mínimo.
O que quero fazer agora é extender as
coisas usando a ideia de concavidade.
Talvez eu esteja pronunciando essa
palavra errado, mas enfim,
pensando sobre concavidade,
podemos começar a olhar
para a segunda derivada mais
do que apenas olhar essa transição

Korean: 
이 점에 접근하면서 증가하고
떠나면서 감소하면
이 점은 최대점이 될 것입니다
비슷하게 여기에서
이 점에 접근하면서 마분값은 음수이고
이것은 함숫값이 감소하는 것을 의미하고
이 점을 떠나면서 미분값이
양수인 것을 알 수 있습니다
우리는 음수인 미분값에서
양수인 미분값으로 갑니다
이것은 함수가 감소하다가 증가하는 것을 의미하고
이것은 꽤 좋은 지표가 됩니다
혹은 이 점은 이것이 임계점이라는 지표가 됩니다
함숫값이 최소가 되는
이제 제가 알고 싶은 것은
볼록성을 이용해서 이 생각을 확장하는 것입니다
 
제가 지금 잘못 발음하고 있는 것을 알고 있습니다
그것은 아마 볼록성입니다
볼록성을 생각하기 위해서는
이차미분을 보는 이 좋습니다
미분값 변화를 보는 것 보다는

Thai: 
เป็นจุดต่ำสุดหรือจุดสูงสุด
ลองคิดถึงสิ่งที่เกิดขึ้น
ในช่วงแรกนี้ ส่วนนี้ของเส้นโค้ง
บนนี้ โดยมันดูเหมือนเส้นโค้งที่กำลัง
คว่ำ มันดูเหมือนตัว A
ที่ไม่มีเส้นขวาง หรือ U กลับหัว
แล้วเราจะคุยกันว่า
เกิดอะไรขึ้นกับส่วนโค้งหงายตัว U นี้
ตลอดช่วงแรกนี้
ตรงนี้ ถ้าเราเริ่มตรงนี้ ความชัน
-- ขอผมทำด้วย - ผมขอทำด้วย
สีเดิมแล้วกัน เพราะมัน
คือสีเดียวกับที่ผมใช้กับอนุพันธ์
ความชันเป็นบวกมาก
แล้วมันเป็นบวกน้อยลง
แล้วมันกลายเป็นบวกน้อยลงอีก
มันกลายเป็น 0
แล้วมันลดลงเรื่อยๆ
ทีนี้ มันเป็นลบนิดหน่อย ลบนิดหน่อย
แล้วมันกลายเป็นลบยิ่งขึ้น
แล้วมันเป็นลบยิ่งขึ้นไปอีก
แล้วมันหยุดลดลงแถวนี้
ความชันหยุดลดตรงแถวนี้
และคุณเห็นว่าในอนุพันธ์

Korean: 
이것이 최소점인지 최대점인지 알기 위해
무슨 일이 일어나는지 봅시다
첫 번째 부분에서
이 곡선에서
호 처럼 생긴 시작부분이 아래에 있는
A처럼 생겼습니다
사이의 직선을 없앤 혹은 뒤집어진 U같이 생겼습니다
이제 생각해봅시다
U처럼 생긴 곡선에서는 무슨 일이 일어나는지
첫번째 구간에서
여기에서 시작하면
기울기는 매우
같은 색으로 합시다
실제 미분의 색이랑 같아서 그렇습니다
기울기는 매우 큰 양수입니다
이것이 점점 작아집니다
점점 더 작아지다가
결국 0이 됩니다
그리곤 계속 감소합니다
그다음 이것은 약간 음수가 되었다가
점점 더 음수가 되었다가
절댓값이 매우 큰 음수가 됩니다
그리곤 이 부근에서 감소하는 것을 멈추는 것처럼 보입니다
기울기가 이 부근에서 감소하는 것을 멈춥니다
그것을 미분에서 볼 수 있습니다

Portuguese: 
para pensar onde isso é
ponto mínimo ou máximo.
Pensemos sobre o que
está acontecendo
nessa primeira região,
nessa parte da curva aqui
onde se parece com um arco
se abrindo para baixo,
onde parece um A sem o traço
ou um U de cabeça para baixo.
E então vamos pensar no que está
acontecendo nessa espécie de U da curva.
Nesse primeiro intervalo,
bem aqui, se começarmos aqui
a inclinação é bastante-- vou fazer na
cor... na verdade vou continuar na mesma
porque é a mesma cor que usei
para essa derivada.
A inclinação é bem positiva.
E se torna menos positiva.
Então se torna ainda menos positiva.
E vai para zero.
E continua decrescendo.
Agora se torna um pouco negativa,
e então ainda mais negativa,
e se torna ainda mais negativa.
E parece que para de decrescer
mais ou menos aqui.
Ela para de decrescer por aqui.
E vemos isso na derivada.

English: 
is a minimum or a maximum point.
So let's think about
what's happening
in this first region,
this part of the curve
up here where it looks like
a arc where it's opening
downward, where it
looks kind of like an A
without the cross beam
or an upside down U.
And then we'll
think about what's
happening in this kind of upward
opening U part of the curve.
So over this first
interval, right
over here, if we start
over here the slope
is very-- actually let me do
it in a-- actually I'll do it
in that same color,
because that's
the same color I used for
the actual derivative.
The slope is very positive.
Then it becomes less positive.
Then it becomes
even less positive.
It eventually gets to 0.
Then it keeps decreasing.
Now it becomes slightly
negative, slightly negative,
then it becomes
even more negative,
then it becomes
even more negative.
And then it looks like it stops
decreasing right around there.
So the slope stops decreasing
right around there.
And you see that
in the derivative.

Bulgarian: 
са точки на минимум или максимум.
Нека да помислим какво се случва
в ето тази област, т.е. тази част от кривата
тук горе, която изглежда като арка, която
е отворена отдолу, и изглежда 
като буквата A,
без пресечната линия, 
или като обърнатa буква U.
Тогава искаме да помислим какво
се случва при тази отворена
 отдолу част U от кривата.
В рамките на този първи интервал, точно
ето тук, ако започнем от тук, то наклонът
е много...Всъщност ще го направя
със същия цвят, защото това
е същият цвят, който използвах 
за производната.
Наклонът е силно положителен.
След това става по-малко 
положителен.
След това става дори 
още по-малко положителен.
Евентуално достига до 0.
След това продължава да намалява.
Сега става слабо отрицателен, 
още повече отрицателен,
а след това става дори 
още по-силно отрицателен.
И дори още повече отрицателен.
Изглежда, че около това място 
спира да намалява.
Наклонът спира да намалява 
точно около тази точка.
Виждаме това на графиката 
на производната.

Chinese: 
whether this is a minimum or maximum point. So
let’s think about what’s happening in this first region. This kind of …this part of
the curve up here where is it looks like an arc where it’s
opening downward. Where it looks kinda like an “A” without the crossbeam or upside
down “U” and then we’ll think about what’s happening in this kind of upward
opening “U”, part of the curve. So over this first
interval right over here, if we start we get this slope is very…is
very ( actually I’ll do it in the same color, exactly the same color that
I used for the actual derivative) the slope is very positive
..slope is very positive. Then it becomes less positive...becomes
less positive…then it becomes even less positive…becomes even less
positive…and eventually gets to zero…eventually gets to zero. Then it keeps
decreasing. Now becomes slightly negative…slightly negative. Then
it becomes even more negative…becomes even more negative…and
then it stops decreasing right around. It looks like it stops decreasing right
around there. So the slope stops decreasing right around there. You see that in the red ,

Czech: 
zda má funkce v daném bodě
lokální minimum nebo maximum.
Zamysleme se nejdřív,
co se děje v této části grafu.
Máme tu část křivky, která vypadá trochu
jako oblouk, jenž se otevírá směrem dolů.
Vypadá to trochu jako A bez čáry
uprostřed nebo jako obrácené U.
Pak se podíváme na to, co se děje v této
části, kde má křivka tvar písmene U.
Když začneme v tomto prvním intervalu,
vidíme, že je sklon velmi...
Udělám to tou samou barvou,
kterou jsem použil pro derivaci.
...sklon je velmi kladný.
Poté je čím dál tím míň kladný,
až se nakonec rovná nule,
načež neustále klesá a stává se
trochu záporným, pak je víc a víc záporný
a nakonec někde
tady přestává klesat.
Sklon funkce tady
přestane klesat.
Vidíme to i
na grafu derivace.

Thai: 
ความชันกำลังลดลง ลดลง ลดลง
ลดลง กระทั่งจุดนั้น แล้วมันก็เริ่มเพิ่มขึ้น
ส่วนทั้งหมดนี่ตรงนี้
ความชันกำลังลดลง
 
แล้วคุณเห็นตรงนี้ เมื่อเราหาอนุพันธ์
อนุพันธ์ตรงนี้ ตลอดช่วงนี้
กำลังลดลง
แล้วเราเห็นด้วยว่า เมื่อเราหาอนุพันธ์อันดับสอง
ถ้าอนุพันธ์กำลังลดลง
นั่นหมายความว่าอนุพันธ์อันดับสอง อนุพันธ์
ของอนุพันธ์ จะเป็นลบ
แล้วเราเห็นว่า มันเป็นเช่นนั้นจริง
ตลอดช่วงทั้งหมดนี้ อนุพันธ์อันดับสอง
เป็นลบจริง
ทีนี้ เกิดอะไรขึ้นเมื่อเราเปลี่ยนผ่าน
ไปยังส่วนหงายของ U ตรงนี้?
ตรงนี้ อนุพันธ์เป็นลบทีเดียว
มันเป็นลบทีเดียวตรงนี้
แต่มันยังเป็นลบ แต่มัน

Portuguese: 
A inclinação está decrescendo
e decrescendo
até esse ponto, e então começa
a ficar crescente.
Em toda essa seção,
a inclinação está decrescendo.
Vemos isso bem aqui quando
pegamos a derivada.
A derivada bem aqui, nesse intervalo
inteiro, está decrescendo.
E também vemos quando temos
a segunda derivada.
Se a derivada é decrescente, significa
que a segunda derivada,
a derivada da derivada, é negativa.
Vemos que esse é mesmo o caso.
Sobre esse intervalo inteiro, a segunda
derivada é, sem dúvida, negativa.
Agora, o que acontece quando mudamos
para essa parte em U da curva?
Bem, aqui a derivada é
razoavelmente negativa.
É razoavelmente negativa por aqui.
Mas ainda continua negativa,
e então se torna menos negativa

Czech: 
Sklon klesá a klesá
až do tohoto bodu,
načež začíná růst.
V celé této části tedy
sklon funkce klesá.
Vidíme to i
v grafu derivace.
Derivace na celém
tomto intervalu klesá.
Co vidíme, když se podíváme
na druhou derivaci?
Když první
derivace klesá,
tak je druhá derivace,
tedy derivace derivace, záporná.
Vidíme, že to tak
opravdu je.
Na celém tomto intervalu je
druhá derivace skutečně záporná.
Co se teď stane, když přejdeme do
části, kde křivka vypadá jako U?
Zde je derivace poměrně záporná,
ale pak se začíná...

Chinese: 
the slope is decreasing…decreasing…decreasing……until that point and
then it starts to increase. So this entire section,
this entire section right over here…
the slope is decreasing. “Slope…
slope is decreasing” and
you see it right over here when we take the derivative, the deri…ative
right over here… the entire, over this entire interval is decreasing.
And we also see that when we take the second derivative. If the derivative is
decreasing that means that the second, the derivative of the derivative is
negative and we see that is indeed the case
over this entire interval. The second derivative, the second
derivative is indeed negative. Now what
happens as we start to transition to this upward opening ”U” part of the curve.
Well here the derivative is reasonably negative,
it’s reasonably negative right there. But then it starts gets…it’s

Korean: 
기울기가 감소하다가
이 점에 닿을 때 까지 감소하고 증가하기 시작합니다
그래서 여기 전 구역에서
기울기는 감소하고 있습니다
 
여기서 미분을 하면
여기서 미분을 하면
여기 구간은
감소하고 있습니다
이차 미분을 하면
미분값이 감소한다는 것은
2차 미분값이
음수라는 것을 의미합니다
이 경우를 보자면
이 구간에서 2차 미분값은
정말 음수입니다
무슨 일이 일어나나요
이 U자 모양 곡선에서?
여기서 미분값은 음수입니다
여기서 미분값은 움수입니다
여기서도 여전히 음수이지만

English: 
The slope is decreasing,
decreasing, decreasing,
decreasing until that point,
and then it starts to increase.
So this entire section
right over here,
the slope is decreasing.
And you see it right over here
when we take the derivative.
The derivative right over
here, over this entire interval
is decreasing.
And we also see that when we
take the second derivative.
If the derivative
is decreasing, that
means that the second
derivative, the derivative
of the derivative, is negative.
And we see that that
is indeed the case.
Over this entire interval,
the second derivative
is indeed negative.
Now what happens as
we start to transition
to this upward opening
U part of the curve?
Well, here the derivative
is reasonably negative.
It's reasonably
negative right there.
But then it's still
negative, but it

Bulgarian: 
Наклонът е намаляващ, 
намаляващ, намаляващ,
докато не достигне до тази точка. 
След това започва да нараства.
Следователно в цялата 
тази част ето тук,
наклонът е намаляващ.
Наклонът е намаляващ.
И това го виждаме ето тук, когато 
наблюдаваме производната.
Производната ето тук, 
в рамките на този интервал,
е намаляваща.
Това се вижда и при втората 
производна.
Ако производната е намаляваща,
това означава, че втората 
производна – т.е. производната
на производната – е отрицателна.
И виждаме действително, 
че случаят е такъв.
В рамките на този интервал 
втората производна
действително е отрицателна.
А какво се случва при преминаването
към тази отворена отгоре крива, 
която изглежда като буквата U?
Тук производната е 
доста отрицателна.
Доста отрицателна е 
на това място.
След това отново е отрицателна,

Chinese: 
still negative but it becomes less negative and less negative
…then it becomes zero,
it becomes zero right over here. And then it becomes more and
more and more positive, and you see that right over here. So over this
entire interval, the slope or the derivative is increasing.
So the slope...slope is
is increasing…the slope is increasing.And you
see this over here, over there the slope is zero. The slope of the derivative is
zero, the slope of the derivative self isn’t changing right this moment and then
…and then you see that the slope is increasing.
And once again we can visualize that on the second derivative, the derivative of the derivative.
If the derivative is increasing that means the derivative of that must be positive.
And it is indeed the case that the derivative is
positive. And we have a word for this downward
opening “U” and this upward opening “U”. we call this
“ concave downwards”

English: 
becomes less negative and less
negative and less negative,
less negative and less
negative, and less negative.
Then it becomes 0.
It becomes 0 right over here.
And then it becomes more
and more and more positive.
And you see that
right over here.
So over this entire interval,
the slope or the derivative
is increasing.
So the slope is increasing.
And you see this over here.
Over here the slope is 0.
The slope of the
derivative is 0.
The derivative itself isn't
changing right at this moment.
And then you see that
the slope is increasing.
And once again, we
can visualize that
on the second derivative, the
derivative of the derivative.
If the derivative
is increasing, that
means the derivative of
that must be positive.
And it is indeed the case that
the derivative is positive.
And we have a word for
this downward opening
U and this upward opening U.
We call this concave downwards.

Portuguese: 
e menos negativa e menos negativa...
E então zero.
Se torna zero bem aqui.
E fica mais e mais positiva.
Vemos isso aqui.
Sobre todo esse intervalo,
a inclinação ou a derivada está crescendo.
A inclinação está crescendo.
E vemos isso aqui.
Aqui a inclinação é zero.
A inclinação da derivada é zero.
A derivada por si só não está
mudando nesse momento.
Vemos que a inclinação
está crescendo.
Novamente, podemos visualizar
a segunda derivada, a derivada
da derivada.
Se a derivada está crescendo, significa
que a derivada deve ser positiva.
E esse é, definitivamente, o caso
onde a derivada é positiva.
Temos uma palavra para o U
de cabeça para baixo e o normal.

Korean: 
점점 덜 음수쪽으로 가다가
점점 덜 음수쪽으로 가다가
0이 됩니다
저기서 0이 됩니다
그리곤 점점 더 커지다가
여기서 볼 수 있듯이
이 전 구간에서 기울기 혹은 미분값이
증가하고 있습니다
기울기가 증가하고 있습니다
여기서는
기울기가 0입니다
미분값이 0입니다
이 순간에는 미분값이 변하지 않습니다
그리고 기울기가 증가하는 것을 볼 수 있습니다
그리고 우리는 저것을 2차 미분을 통해 시각화할 수 있습니다
미분의 미분
미분이 증가하고 있다는 것은
미분의 미분이 양수라는 것을 의미합니다
미분의 미분이 정말 양수입니다
그리고 뒤집어진 U와 그냥 U자 모양을 칭하는
용어가 있습니다.
우리는 이것을 위로 볼록하다고 말합니다

Bulgarian: 
но става все по-малко, 
по-малко и по-малко отрицателна,
По-малко, по-малко и 
по-малко отрицателна,
След това достига до 0.
Ето в тази точка става 0.
След това става все повече 
и повече положителна.
Това се вижда и ето тук.
Следователно в рамките на целия този
 интервал наклонът, или производната,
е нарастваща.
Наклонът е нарастващ.
Това се вижда ето тук.
В тази точка наклонът е 0.
Наклонът на производната е 0.
Самата производна не се променя
 в този момент.
След това се вижда, че 
наклонът нараства.
Още веднъж, може да онагледим това
на графиката на втората производна, 
т.е. производната на производната.
Ако производната е нарастваща,
това означава, че нейната производна
следва да е положителна.
И действително се получава, че 
производната е положителна.
Имаме дума за тази отворена отдолу
крива с формата на U, и отворена отгоре крива 
с формата на U. Наричаме това вдлъбната функция.

Czech: 
Sice je pořád záporná, ale stává se
čím dál tím méně zápornou,
až se nakonec
v tomto bodě rovná nule,
načež je stále více kladná,
jak je tady vidět.
Na celém tomhle intervalu
sklon funkce, tedy její derivace, roste.
Sklon roste.
Můžeme zde vidět,
že v tomhle bodě je sklon nulový.
Sklon derivace je nula, samotná
derivace se v tuto chvíli nemění.
Následně vidíme,
že sklon roste.
Opět se můžeme podívat, jak bude vypadat
druhá derivace, tedy derivace derivace.
Když je první derivace rostoucí,
druhá derivace musí být kladná.
Druhá derivace je
zde skutečně kladná.
Existuje speciální název
pro části grafu,
které vypadají jako převrácené
nebo normální písmeno U.

Thai: 
เป็นลบน้อยลง ลบน้อยลง และลบน้อยลง
ลบน้อยลง แล้วก็ลบน้อยลง ลบน้อยลง
แล้วมันกลายเป็น 0
มันกลายเป็น 0 ตรงนี้
แล้วมันก็กลายเป็นบวกมากขึ้น มากขึ้น และมากขึ้น
แล้วคุณเห็นตรงนี้
ตลอดช่วงทั้งหมดนี้ ความชันหรืออนุพันธ์
กำลังเพิ่มขึ้น
ความชันจะเพิ่มขึ้น
และคุณเห็นมันตรงนี้
ตรงนี้ ความชันเป็น 0
ความชันของอนุพันธ์เป็น 0
อนุพันธ์เองไม่ได้เปลี่ยนในขณะนี้
แล้วคุณเห็นว่าความชันกำลังเพิ่มขึ้น
เหมือนเดิม เรามองภาพมัน
ในอนุพันธ์อันดับสอง อนุพันธ์ของอนุพันธ์
ถ้าอนุพันธ์เพิ่มขึ้น
นั่นหมายความว่า อนุพันธ์ของมันต้องเป็นบวก
และมันเป็นจริงที่อนุพันธ์เป็นบวก
แล้วเรามีคำเรียก U คว่ำนี้
และ U หงายนี้ เราเรียนมันว่าเว้าลง

Portuguese: 
Chamamos de côncavo para baixo.
Vou ser mais claro.
Côncavo para baixo.
E chamamos esse de côncavo para cima.
Vamos revisar como podemos identificar
intervalos côncavos para baixo e
intervalos côncavos para cima.
Se estamos querendo côncavo para baixo,
vemos bastante coisa.
Vemos que a inclinação está decrescendo,
que é outra maneira de dizer que
f linha de x está decrescendo.
O que é outra maneira de dizer que
a segunda derivada deve ser negativa.
Se a primeira derivada está decrescendo,

English: 
Let me make this clear.
Concave downwards.
And we call this
concave upwards.
So let's review
how we can identify
concave downward intervals
and concave upwards intervals.
So if we're talking
about concave downwards,
we see several things.
We see that the
slope is decreasing.
Which is another
way of saying that f
prime of x is decreasing.
Which is another way of saying
that the second derivative must
be negative.
If the first derivative
is decreasing,

Chinese: 
(let me make this clear)…
concave downwards.
And we call this “ concave upwards”… concave
upwards. So let’s review how we can identify concave
downwards intervals and upwards intervals. So we are talking about concave
downwards…”concave downwards”.
We see several things,
we see that the slope is decreasing, the slope is
is decreasing.“The slope
is decreasing” which is another way of saying,
which is another way of saying that f’(x)
is decreasing.
decreasing. Which is another way of saying that the
second derivative must be negative. If the first derivative is decreasing, the second

Korean: 
 
분명히 하도록 하겠습니다
위로 볼록
그리고 우리는 이것을 위로 아래로 볼록이라고 부릅니다
 
복습해봅시다 우리가 어떻게
아래로 볼록인 구간과 위로 볼록인 구간을 확인할 수 있는지
우리가 위로 볼록을 애기할 때
우리는 몇 가지 것들을 볼 수 있습니다
기울기가 감소하는 것을 알 수 있고
 
다른 말로 f'(x)가 감소하고 있습니다
 
 
또 다른 말로 2차 미분값이
음수입니다
1차 미분값이 감소하면

Czech: 
Řekneme, že v této části
je funkce konkávní
a že zde je funkce
konvexní.
Pojďme si zopakovat, jak určíme intervaly,
na kterých je funkce konkávní či konvexní.
Když máme interval, na kterém je
funkce konkávní, tak vidíme několik věcí.
Vidíme, že sklon
funkce je klesající,
což je totéž jako říci,
že derivace funkce f je klesající,
a to jinak řečeno znamená,
že druhá derivace musí být záporná.

Thai: 
 
ขอผมบอกให้ชัดนะ
เว้าลง
และเราเรียกส่วนนี้ว่า เว้าขึ้น
 
ลองทบทวนกันว่าเราระบุ
ช่วงเว้าลง กับช่วงเว้าขึ้นว่าอย่างไร
ถ้าเราพูดถึงการเว้าลง
เราเห็นหลายอย่าง
เราเห็นว่าความชันกำลังลดลง
 
ซึ่งก็คือวิธีบอกว่า f
ไพรม์ของ x กำลังลดลง
 
ซึ่งเป็นวิธีบอกอีกอย่างว่า อนุพันธ์อันดับสองต้อง
เป็นลบ
ถ้าอนุพันธ์อันดับหนึ่งลดลง

Bulgarian: 
Вдлъбната функция.
Нека да го изясня.
Вдлъбната функция.
А това наричаме изпъкнала функция.
Изпъкнала функция.
Нека да си припомним 
как може да открием
интервали на вдлъбнатост 
и интервали на изпъкналост.
Ако говорим за вдлъбната функция,
виждаме няколко неща.
Виждаме, че наклонът е намаляващ.
Наклонът е намаляващ.
Което е просто друг начин да кажем,
 че f' от х е намаляваща.
Което е друг начин да се каже, 
че втората производна
трябва да е отрицателна.
Ако първата производна намалява,

Thai: 
อนุพันธ์อันดับสองต้องเป็นลบ
ซึ่งเป็นวิธีบอกอีกอย่างว่า อนุพันธ์อันดับสอง
ตลอดช่วงนั้นต้องเป็นลบ
ถ้าคุณมีอนุพันธ์อันดับสองเป็นลบ
แล้วคุณจะมีช่วงเว้าลง
เช่นเดียวกัน -- ผมมีปัญหาการพูด
คำนั้นเหลือเกิน -- ลองคิดถึงการเว้าขึ้น
โดยคุณมี U หงาย เว้าขึ้น
ในช่วงเหล่านี้ ความชันจะเพิ่มขึ้น
เรามีความชันเป็นลบ ลบน้อยลง ลบน้อยลง 0
บวก บวกขึ้น บวกขึ้น บวกขึ้นอีก
ความชันจึงเพิ่มขึ้น
 
ซึ่งหมายความว่าอนุพันธ์ของฟังก์ชันกำลังเพิ่มขึ้น
และคุณเห็นมันตรงนี้
อนุพันธ์นี้มีค่าเพิ่มขึ้น

Korean: 
2차 미분값은 반드시 음수입니다
다른 말로 2차 미분값이
저 구간에서 반드시 음수입니다
그래서 음의 2차 미분값을 가지고 있으면
위로 볼록인 구간입니다
비슷하게
발음하기가 힘듭니다
아래로 볼록에 대해 생각해 봅시다
U자 모양의 구간에서 아래로 볼록입니다
이 구간에서 기울기는 증가하고 있습니다
우리는 음의 기울기, 덜 음의 기울기
0이 됩니다
양의 기울기에서 점점 커집니다
그래서 기울기가 증가하고 있습니다
 
이것은 함수의 미분값이 증가한다는 것을 의미합니다
저기서 볼 수 있듯이
이 미분이 증가하고 있습니다

Chinese: 
the second derivative must be negative. Which is another way
of saying that the second derivative of that interval must
be… must be negative. So if you have
negative second derivative, then you are in a concave
downward interval. Similarly…similarly
(I have trouble saying that word), let’s think about concave upwards,
where you have an upward opening “U”. Concave upwards.
In these intervals, the slope is increasing,
we have negative slope, less negative, less negative…zero, positive, more positive, more
positive…even more positive. So slope...slope
is increasing. "Slope is
increasing which means
that the derivative of the function is
increasing. And you see that right over
here, this derivative is increasing in value,
which means that the second derivative，the second derivative

English: 
the second derivative
must be negative,
which is another way of saying
that the second derivative
over that interval
must be negative.
So if you have a negative
second derivative,
then you are in a concave
downward interval.
Similarly-- I have
trouble saying
that word-- let's think
about concave upwards,
where you have an upward
opening U. Concave upwards.
In these intervals, the
slope is increasing.
We have a negative slope, less
negative, less negative, 0,
positive, more positive, more
positive, even more positive.
So slope is increasing.
Which means that the derivative
of the function is increasing.
And you see that
right over here.
This derivative is
increasing in value,

Czech: 
Jestliže první derivace klesá,
druhá derivace musí být záporná.
Tohle je tedy
totéž jako říci,
že druhá derivace musí být
na daném intervalu záporná.
Je-li tedy
druhá derivace záporná,
tak je funkce na
daném intervalu konkávní.
Obdobně...
Dělá mi potíže
to slovo vyslovit.
...obdobně se podívejme na interval,
na němž je funkce konvexní,
tedy kde má graf
funkce tvar U.
Konvexní.
Na těchto intervalech
je sklon rostoucí.
Sklon je záporný, pak méně záporný,
ještě méně záporný, nulový, kladný,
kladnější,
ještě kladnější...
Sklon je
tedy rostoucí,
což znamená, že derivace
funkce je rostoucí.
To můžeme
vidět zde.
Hodnota derivace roste.
Tohle pak znamená,

Bulgarian: 
то втората производна 
трябва да е отрицателна.
А това е друг начин да се каже, 
че втората производна
в рамките на този интервал 
трябва да бъде отрицателна.
Тоест, ако имаш отрицателна 
втора производна,
то тогава се намираш 
в интервал на вдлъбнатост.
Аналогично – имам трудности
 с изговарянето
на тази дума - нека да помислим 
за изпъкналост,
или когато имаш отворена отгоре 
крива U. Изпъкнала функция.
В тези интервали наклонът
 е нарастващ.
Имаме отрицателен наклон, по-малко 
отрицателен, по-малко отрицателен, 0,
положителен, повече положителен, повече 
положителен, още повече положителен.
Следователно наклонът е нарастващ.
Наклонът е нарастващ.
Което означава, че производната 
на функцията е нарастваща.
А това се вижда ето тук.
Производната е нарастваща,

Portuguese: 
a segunda derivada deve ser negativa,
o que é outra forma de dizer
que a segunda derivada
sobre esse intervalo deve ser negativa.
Se temos uma segunda
derivada negativa,
estamos num intervalo
côncavo para baixo.
Similarmente-- tenho dificuldade
de pronunciar essa palavra--
vamos pensar sobre côncavo para cima,
onde temos uma abertura em U para cima.
Côncavo para cima.
Nesses intervalos, a inclinação
está crescendo.
Temos uma inclinação negativa, menos
negativa... zero,
positiva, mais positiva,
ainda mais positiva.
A inclinação está crescendo.
O que significa que a derivada
da função está aumentando.
E vemos isso bem aqui.
A derivada aumenta em valor,

Portuguese: 
o que significa que a segunda derivada
sobre um intervalo
onde é côncavo para cima
deve ser maior que zero.
Se a segunda derivada é maior que zero,
então a primeira derivada está
crescendo, e significa que
a inclinação está crescendo.
Estamos num intervalo côncavo para cima.
Dadas essas definições que obtivemos
de côncavos para baixo e para cima,
podemos chegar a uma outra maneira
de identificar quando um ponto crítico
é um ponto mínimo ou ponto máximo?
Bem, se temos um ponto máximo,
se temos um ponto crítico onde
a função é côncava para baixo,
estaremos em um ponto máximo.
Côncavo para baixo-- sendo mais claros,
significa que se abre para baixo assim.
Quando estamos falando
de um ponto crítico,
se assumirmos que é côncavo
para baixo aqui,
estamos assumindo diferenciabilidade
nesse intervalo.
O ponto crítico será aquele
que a inclinação é zero.
Será esse ponto aqui.

Chinese: 
over the interval where we are concave upwards must be greater than zero,
the second derivative is greater than zero that means the first derivative is increasing,
which means that the slope is increasing. We are in a concave upward,
we are in a concave upward interval. Now,
given all these definitions we’ve just given for concave downwards
and concave upwards interval, can we come out with another way of indentifying whether a critical point
is a minimum point or maximum point.
Well, if you have a maximum point, if you have a critical point where the
function...where the function is concave downwards,
then it going to be a maximum point."Concave downwards". Let’s just be clear here,
means that it’s opening down like this
and we are talking about a critical point. If we’re assuming it’s concave downwards
over here, we’re assuming differentiability over this interval and so the critical point
is gonna be one where the slope is zero, so it’s gonna be that point
right over there. So if you have a concave upwards and you have a point where

Thai: 
ซึ่งหมายความว่าอนุพันธ์อันดับสองในช่วง
โดยเราเหว้าขึ้นต้องมากกว่า 0
ถ้าอนุพันธ์อันดับสองมากกว่า 0
นั่นหมายความว่าอนุพันธ์อันดับหนึ่ง
เพิ่มขึ้น ซึ่งหมายความว่าความชันกำลังเพิ่มขึ้น
เราจะอยู่ในช่วงเว้าขึ้น
ทีนี้ จากนิยามทั้งหมดนี้ที่เราเพิ่ม
กำหนดสำหรับการเว้าลงและเว้าขึ้น
เราหาวิธีบอก
ว่าจุดวิกฤตเป็นจุดต่ำสุด
หรือจุดสูงสุดได้ไหม?
ถ้าคุณมีจุดสูงสุด
ถ้าคุณมีจุดวิกฤตที่ฟังก์ชัน
เว้าลง แล้วคุณจะอยู่ที่จุดสูงสุด
เว้าลง ขอบอกให้ชัดตรงนี้นะ
หมายความว่ามันคว่ำอย่างนี้
และเมื่อเราพูดถึงจุดวิกฤต
ถ้าเราสมมุติว่ามันเว้าลงตรงนี้
เราสมมุติว่าเราหาอนุพันธ์ได้ตลอดช่วงนี้
แล้วจุดวิกฤตจะ
เป็นจุดที่ความชันเป็น 0
มันจะเป็นจุดนั่นตรงนั้น
ถ้าคุณเว้าลง และคุณ

Korean: 
이것은 아래로 볼록인 구간에서
2차 미분값이 0보다 크다는 것을 의미합니다
2차 미분값이 0보다 크면
그것은 1차 미분값이
즉 기울기가 증가하고 있다는 것을 의미합니다
우리는 아래로 볼록인 구간에 있습니다
우리가 위로 볼록과 아래로 볼록에 대해
지금까지 정의한 것을 생각해봅시다
임계점이 최소점인지 최대점인지
알 수 있는 다른 방법에 대해
생각할 수 있나요?
만약 최대점이면
만약에 임계점이 위로 볼록인 구간에 있으면
그 임계점은 최대점이 될 것입니다
위로 볼록에 대해 분명히 합시다
이렇게 생긴 구간을 의미합니다
그리고 우리는 임계점에대 말하고 있습니다
우리가 여기를 위로 볼록이라고 가정하면
이 구간에서 미분 가능하다고 가정합니다
그래서 임계점은
기울기가 0인 점 입니다
그래서 저 점이 될 것입니다
여기서 위로 볼록이라면

Czech: 
že na intervalu, kde je funkce konvexní,
musí být druhá derivace větší než nula.
Když je druhá derivace větší než nula,
tak je první derivace rostoucí,
a tudíž i sklon
funkce je rostoucí.
Funkce je na daném
intervalu konvexní.
Když už jsme si teď zadefinovali,
kdy je funkce konkávní a konvexní,
dokážeme vymyslet
jiný způsob, jak zjistit,
zda má funkce v stacionárním bodě
lokální minimum nebo maximum?
Když má funkce
lokální maximum...
Máme-li stacionární bod,
okolo něhož je funkce konkávní,
tak jde o bod
lokálního maxima.
Funkce musí
být konkávní.
Ještě to
ujasním.
Konkávní znamená,
že graf se takhle otevírá dolů.
Když máme
stacionární bod...
Předpokládejme, že funkce je konkávní a
diferencovatelná na celém tomto intervalu.
Ve stacionárním bodě
bude sklon roven nule,
takže to je
tento bod.

Bulgarian: 
което означава, че втората производна
 в рамките на този интервал –
където функцията е изпъкнала –
 трябва да е по-голяма от 0.
Ако втората производна 
е по-голяма от 0,
това означава, че първата производна
е нарастваща, което означава, 
че наклонът е нарастващ.
Намираме се в интервал 
на изпъкналост.
Като вземем предвид всички тези 
дефиниции –
които току-що дадохме 
за вдлъбнатост и изпъкналост –
можем ли да намерим 
друг начин за определяне
дали една критична точка 
е точка на минимум
или точка на максимум?
Ако имаш точка на максимум,
т.е. критична точка, където функцията
е вдлъбната, тогава ще се намираш
 в точка на максимум.
Вдлъбната функция, нека да го изясним,
означава, че кривата
се отваря отдолу като ето това.
А когато става дума за критична точка,
ако предполагаме, че е 
вдлъбната ето тук,
предполагаме, че функцията е 
диференцируема в този интервал.
Тогава критичната точка ще бъде 
там, къде наклонът е равен на 0.
Следователно ще бъде 
ето тази точка тук.
Ако функцията е вдлъбната и имаш

English: 
which means that the second
derivative over an interval
where we are concave upwards
must be greater than 0.
If the second derivative
is greater than 0,
that means that the
first derivative
is increasing, which means
that the slope is increasing.
We are in a concave
upward interval.
Now given all of these
definitions that we've just
given for concave downwards
and concave upwards,
can we come up with
another way of identifying
whether a critical
point is a minimum point
or a maximum point?
Well, if you have
a maximum point,
if you have a critical
point where the function is
concave downwards, then you're
going to be at a maximum point.
Concave downwards, let's
just be clear here,
means that it's
opening down like this.
And when we're talking
about a critical point,
if we're assuming it's
concave downwards over here,
we're assuming differentiability
over this interval.
And so the critical
point is going
to be one where the slope is 0.
So it's going to be that
point right over there.
So if you're concave
downwards and you

Bulgarian: 
точка, където f' от а, например, 
е равно на 0,
то функцията има 
максимум в точката а.
Mаксимум в точката а.
Аналогично, ако функцията е изпъкнала,
това означава, че функцията 
изглежда като нещо такова.
Ако бяхме намерили 
критична точка, т.е. такава,
при която функцията може 
да не е дефинирана,
но предположим, че първата производна
и втората производна е дефинирана тук,
тогава критичната точка 
ще бъде такава,
при която първата производна
ще бъде равна на 0.
Следователно f' от а е равно на 0.
А ако f' от а е равно на 0,
и функцията е изпъкнала в интервала
около а, и ако втората производна 
е по-голяма от 0,
то е достатъчно ясно –
 ето тук се вижда –
че в точка а има точка на минимум.

Portuguese: 
Se é côncavo para baixo e temos
um ponto onde f linha de,
digamos, a é igual a zero,
temos um ponto máximo em a.
E similarmente, se é côncavo para cima,
significa que a nossa função
se parece com algo assim.
Se achamos um ponto, obviamente
um ponto crítico seria aonde
a função não é definida, mas se estamos
assumindo que nossa primeira derivada
e segunda derivada é definida aqui,
o ponto crítico será aquele
onde a primeira derivada será zero.
f linha de a é igual a zero.
E se f linha de a é igual a zero
e côncavo para cima no
intervalo ao redor de a -
se a segunda derivada
é maior que zero,
é bem claro, podemos ver aqui,
que estamos lidando com
um ponto mínimo em a.
[Legendado por Miguel Infante]
[Revisado por Pilar Dib]

Korean: 
그리고 f'(x)가 0이 되는 점에서는
그 점을 a라고 합시다
그러면 a에서 최댓값을 가집니다
 
그리고 비슷하게 아래로 볼록이면
함수가 이렇게 생긴 것을 의미합니다
그리고 임계점을 찾으면
함수가 정의 되지 않은 부분일 수도 있습니다
하지만 우리가 1차 미분과 2차 미분이
정의된다고 가정하면
임계점은
첫번재 미분값이 0이되는 점일 것입니다
f'(a)=0입니다
만약 f'(a)=0이면
그리고 a주변에서 아래로 볼록이면
즉 2차 미분값이 0보다 크면
꽤 분명하게 볼 수 있듯이
a에서 최소점입니다

Chinese: 
f’(a) = 0
then we have a maximum point at a.
And similarly if we are a concave upwards
that means that our function looks something like this and if we
found the point. Obviously a critical point could also be where the function is not
defined. But if we are assuming that our first derivative and second derivative is
defined here then the critical point is going to be one where the first derivative is
going to be zero, so f’(a)
f’(a)= 0.If f’(a)= 0
and if we are concave upwards and the interval around
a, so the second derivative is greater than zero, then it’s pretty
clear you see here that we are dealing with… we are dealing with a minimum,
a minimum point at a .

Czech: 
Když máme konkávní funkci a bod ‚a‘,
ve kterém je derivace rovna nula,
tak je bod ‚a‘
bodem lokálního maxima.
Podobně to platí i tehdy,
když je funkce konvexní,
což znamená,
že její graf vypadá nějak takto.
Pokud jsme našli
nějaký bod...
Stacionární bod může být samozřejmě také
bod, ve kterém derivace není definovaná,
ale když předpokládáme,
že první i druhá derivace jsou definované,
tak bude stacionárním bodem ten bod,
ve kterém je první derivace rovna nule,
neboli pro který platí,
že f s čárkou v bodě ‚a‘ se rovná nule.
Když je f s čárkou
v bodě ‚a‘ rovno nule
a funkce je na intervalu
okolo bodu ‚a‘ konvexní,
tedy když je
druhá derivace větší než 0,
tak je celkem zřejmé,
že se jedná o lokální minimum,
tedy že funkce má v bodě ‚a‘
svoje lokální minimum.

Thai: 
มีจุดที่ f ไพรม์ของ อย่างเช่น a เท่ากับ 0
แล้วเราจะได้จุดสูงสุดที่ a
 
เช่นเดียวกัน ถ้าเราเว้าขึ้น
นั่นหมายความว่าฟังก์ชันของเราเป็นแบบนี้
และถ้าเราเจอจุด แน่นอน จุดวิกฤต
จะเป็นตำแหน่งที่ฟังก์ชันไม่นิยาม
แต่ถ้าเราสมมุติว่าอนุพันธ์แรกของเรา
และอนุพันธ์อันดับสองนิยามตรงนี้
แล้วจุดวิกฤตจะเป็นจุด
ที่อนุพันธ์อันดับหนึ่งเท่ากับ 0
f ไพรม์ของ a เท่ากับ 0
และถ้า f ไพรม์ของ a เท่ากับ 0
และถ้าเราเว้าขึ้นในช่วง
รอบ a ถ้าอนุพันธ์อันดับสองมากกว่า 0
มันจะเห็นได้ชัด คุณเห็นตรงนี้
ว่าเราจะได้จุดต่ำสุดที่ a

English: 
have a point where f prime of,
let's say, a is equal to 0,
then we have a
maximum point at a.
And similarly, if
we're concave upwards,
that means that our function
looks something like this.
And if we found a point,
obviously a critical point
could also be where the
function is not defined,
but if we're assuming
that our first derivative
and second derivative
is defined here,
then the critical point
is going to be one
where the first derivative
is going to be 0.
So f prime of a is equal to 0.
And if f prime of
a is equal to 0
and if we're concave
upwards in the interval
around a, so if the second
derivative is greater than 0,
then it's pretty
clear, you see here,
that we are dealing with
a minimum point at a.
