泰勒公式外文翻译

Ｔa ｙl ｏr ＇s Ｆormu ｌa an ｄ the St ｕdy ｏf Ext ｒema

1. Ｔay ｌｏr's Ｆorm ｕｌa for M ａpping ｓ

Th ｅｏre ｍ 1. If ａ mapping Y U f →: from a ｎeigh ｂo ｒho ｏｄ ()x U U = of a point x in a no ｒm ｅd ｓｐａce X into a no ｒmed spac ｅ Y ha ｓ d ｅｒiva ｔiv ｅs up ｔo o ｒde ｒ n -１ in ｃlus ｉve ｉｎ U an ｄ h ａｓ ａn n-th ｏrde ｒ deriv ａti ｖe ()()x f n at t ｈe p ｏint x, ｔhe ｎ

()()()()()⎪⎭⎫ ⎝⎛++++=+n n n h o h x f n h x f x f h x f !1,

(１）

as 0→h ．

Equ ａli ｔy （１) i ｓ one of the ｖariet ｉe ｓ ｏf Ｔa ｙlor's fo ｒmu ｌａ, writte ｎ here for ra ｔｈer ge ｎｅral c ｌas ｓｅs of ma ｐpings.

Proof ． We p ｒｏv ｅ T ａy ｌｏr's ｆo ｒmu ｌa ｂｙ ｉnd ｕｃｔi ｏn. For 1=n it is tru ｅ ｂｙ d ｅｆinition ｏf ()x f ,.

Ａs ｓume form ｕl ａ (1) is ｔrue for ｓo ｍe N n ∈-1.

Ｔｈen b ｙ the mean －val ｕｅ th ｅo ｒｅm, fo ｒm ｕla （12) o ｆ Sec ｔ. 10.5, ａnd the ｉn ｄuc ｔion h ｙｐot ｈesis ， we obtain.

()()()()()()()()()()()()()⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛=⎪⎪⎭

⎫ ⎝⎛-+++-+≤⎪⎭

⎫ ⎝⎛+++-+--<

h x f n h x f x f h x f h x f n h f x f h x f 11,,,,10!11sup !1x θθθθθ ，

ａs 0→h .

W ｅ s ｈall ｎot take t ｈｅ t ｉｍe h ｅre t ｏ di ｓcuss o ｔh ｅr v ｅr ｓi ｏns of Ｔayl ｏr ＇ｓ f ｏｒmu ｌａ, ｗhic ｈ are so ｍeti ｍes quite useful ． Ｔhey ｗer ｅ discu ｓse ｄ earlier in ｄｅｔail ｆｏr ｎume ｒica ｌ fun ｃti ｏｎs ． At thi ｓ point w ｅ le ａv ｅ it ｔｏ th ｅ reader to derive ｔhem (s ｅe, for e ｘample ， Pro ｂlem 1 below)．

2. Ｍeth ｏd ｓ of St ｕdying Ｉｎte ｒｉor Ｅxt ｒema

Usin ｇ Ta ｙlor'ｓ formul ａ, we sha ｌｌ ｅｘhib ｉt nece ｓsar ｙ condi ｔio ｎs a ｎｄ a ｌso suffic ｉe ｎt c ｏndi ｔions for a ｎ ｉｎｔe ｒior l ｏcal ex ｔre ｍum ｏf r ｅａl －ｖal ｕe ｄ fun ｃtions def ｉned on ａn ｏp ｅn su ｂse ｔ ｏf ａ nor ｍed sp ａce. Ａs we sha ｌl ｓee ， ｔｈｅse cond ｉt ｉo ｎs are an ａｌogous to t ｈe diffe ｒｅnti ａl conditi ｏｎs alr ｅady known t ｏ ｕs fo ｒ ａｎ ex ｔr ｅmum o ｆ a ｒeal －ｖal ｕe ｄ ｆunc ｔion of a ｒe ａｌ ｖariabl ｅ.

Ｔｈeorem 2. Let R U f →: be ａ ｒeal-v ａl ｕed functio ｎ d ｅｆｉned o ｎ an

o ｐｅｎ set Ｕ in a ｎorm ｅd ｓpa ｃｅ X a ｎd ｈaving ｃontinuous derivatives up to ｏrder 11≥-k inclu ｓiv ｅ ｉn a neig ｈb ｏｒh ｏod of a point U x ∈ an ｄ a d ｅｒｉvative

()()x f k of order ｋ at th ｅ p ｏi ｎt ｘ i ｔself. If ()()()0,,01,==-x f x f k an ｄ ()()0≠x f k , ｔhen for x ｔo be ａn e ｘt ｒｅm ｕm of ｔｈｅ fu ｎｃt ｉｏn ｆ ｉt ｉs:

n ｅces ｓary ｔhat k be ev ｅn and th ａt the form

()()k k h x f b ｅ s ｅmid ｅfini ｔｅ,

a ｎd ｓｕfficient ｔh ａt the v ａlues o ｆ t ｈe ｆor ｍ

()()k k h x f ｏn th ｅ ｕnit s ｐｈere 1=h be b ｏund ｅｄ away ｆr ｏm ze ｒo; ｍo ｒeov ｅｒ, x i ｓ a lo ｃal ｍinimum ｉｆ the i ｎｅquali ｔi ｅs

()()0>≥δk k h x f ,

h ｏｌd o ｎ that ｓｐｈere, and a ｌo ｃal ma ｘimum if

()()0<≤δk k h x f ，

Ｐroof. For the pr ｏof we con ｓi ｄe ｒ the Taylor ｅｘｐansio ｎ (1) of f i ｎ a ｎeigh ｂo ｒhood of x ． Ｔh ｅ ass ｕmptions e ｎable u ｓ ｔo w ｒite

()()()()()k k k h h h x f k x f h x f α+=-+!1

where ()h α is a re ａl-valued f ｕnc ｔio ｎ, and ()0→h α as 0→h ．

W ｅ fi ｒs ｔ prove th ｅ neces ｓａｒｙ co ｎditions.

Since ()()0≠x f k , th ｅr ｅ ｅxist ｓ a v ｅctor 00≠h on w ｈic ｈ()()00≠k k h x f . The ｎ for values of th ｅ real pa ｒam ｅｔer ｔ suffici ｅntly close to ｚero ，

()()()()()()k k k th th th x f k x f th x f 0000!1α+=-+

()()()k

k k k t h th h x f k ⎪⎭⎫ ⎝⎛+=000!1α

a ｎd the ex ｐr ｅｓsion in the outer pa ｒent ｈｅｓes h ａs the same sign a ｓ()()k k h x f 0.

For ｘ ｔo be an e ｘtre ｍum i ｔ i ｓ neces ｓary for the left-hand ｓid ｅ (an ｄ h ｅnce ａlso ｔｈｅ ｒight-h ａｎd side) ｏf this last e ｑuality to be of co ｎｓｔan ｔ sign ｗｈｅｎ t c ｈan ｇes sig ｎ. B ｕt th ｉｓ is pos ｓible only ｉf k is ev ｅn ．

Thi ｓ reaso ｎing sho ｗs that if ｘ ｉｓ an ex ｔre ｍum, ｔhen the sign o ｆ the diffe ｒen ｃe ()()x f th x f -+0 ｉs th ｅ ｓａme a ｓ th ａt of ()()k k h x f 0 fo ｒ ｓuffi ｃi ｅ

ｎt ｌy sma ｌl t; hen ｃe in that ca ｓｅ th ｅｒｅ ｃａｎn ｏt be two vect ｏrs

0h , 1h