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Îö³íèìî n ( )<br />
n+ 1 θx<br />
n+<br />
1<br />
x e x<br />
x<br />
R x = ≤ e<br />
1! 1!<br />
, àëå<br />
( n + ) ( n + )<br />
ïðè áóäü-ÿêîìó ñê³í÷åííîìó õ, ³ òîìó Rn<br />
( x)<br />
n+<br />
1<br />
x<br />
lim = 0<br />
n→∞<br />
1!<br />
( n + )<br />
lim = 0 . Îòæå, çà<br />
n→∞<br />
ôîðìóëîþ (7.15.18) çíà÷åííÿ å õ ìîæíà îá÷èñëèòè ç áóäüÿêèì<br />
ñòåïåíåì òî÷íîñò³.<br />
Âðàõîâóþ÷è ôîðìóëè (7.9.9) – (7.9.10), íå âàæêî çàïèñàòè<br />
ùå äâ³ äóæå âàæëèâ³ ôîðìóëè Ìàêëîðåíà:<br />
π<br />
sin( θ x+ (2n+<br />
1) )<br />
x x x<br />
sin x = x− + + ... + ( − 1) +<br />
2<br />
x<br />
3! 5! (2n− 1)! (2n+<br />
1)!<br />
3 5 2n−1<br />
n− 1 2n+<br />
1<br />
( x n)<br />
x x − x cos θ +π<br />
cos x = 1 − + + ... + ( − 1)<br />
+<br />
2! 4! (2n−<br />
2)! (2 n)!<br />
2 4 2n−2<br />
n 1 2n<br />
x<br />
; (7.15.19)<br />
. (7.15.20)<br />
7.16. ÍÅÎÁÕ²ÄͲ ÒÀ ÄÎÑÒÀÒͲ ÓÌÎÂÈ<br />
²ÑÍÓÂÀÍÍß ÅÊÑÒÐÅÌÓÌÓ ÔÓÍÊÖ²¯<br />
7.16.1. Íåîáõ³äíà óìîâà ³ñíóâàííÿ åêñòðåìóìó<br />
ôóíêö³¿<br />
Òåîðåìà Ôåðìà ÷àñòêîâî âêàçóº íà íåîáõ³äíó óìîâó ³ñíóâàííÿ<br />
åêñòðåìóìó ôóíêö³¿. Á³ëüø çàãàëüíó òåîðåìó äîâåäåìî<br />
â öüîìó ïóíêò³.<br />
Òåîðåìà 7.16.1. ßêùî ôóíêö³ÿ y = f(x) â òî÷ö³ õ 0 ìàº<br />
ëîêàëüíèé åêñòðåìóì, òî âîíà â ö³é òî÷ö³ àáî íå äèôåðåíö³éîâíà,<br />
àáî ìຠïîõ³äíó, ÿêà äîð³âíþº íóëþ.<br />
Äîâåäåííÿ. Íåõàé õ 0 òî÷êà ëîêàëüíîãî åêñòðåìóìó ³<br />
ôóíêö³ÿ f(x) â ö³é òî÷ö³ äèôåðåíö³éîâíà. Îñê³ëüêè çíà÷åííÿ<br />
f(x 0 ) º íàéá³ëüøå àáî íàéìåíøå â îêîë³ òî÷êè õ 0 , òî çà<br />
òåîðåìîþ Ôåðìà f′(x 0 )=0.<br />
Âèïàäîê ôóíêö³¿, íå äèôåðåíö³éîâí³é ó òî÷ö³ åêñòðåìóìó,<br />
ïðî³ëþñòðóºìî íà ïðèêëàä³. ijéñíî ôóíêö³ÿ y =|x| â òî÷ö³<br />
õ = 0 ìຠì³í³ìóì, àëå íå äèôåðåíö³éîâíà â í³é (äèâ.<br />
ïðèêë. 7.3.2). Òåîðåìà 7.16.1 ìຠïðîñòèé ãåîìåòðè÷íèé<br />
çì³ñò: ó òî÷ö³ ãðàô³êà ôóíêö³¿, ÿêà â³äïîâ³äຠòî÷ö³ ëîêàëüíîãî<br />
åêñòðåìóìó, äîòè÷íà àáî ïàðàëåëüíà îñÿì êîîðäèíàò,<br />
àáî íå ³ñíóº.<br />
Äîâåäåíà óìîâà åêñòðåìóìó º íåîáõ³äíîþ, àëå íå äîñòàòíüîþ.<br />
Íàïðèêëàä, ôóíêö³ÿ ó = õ 3 â òî÷ö³ õ = 0 ìຠïîõ³äíó,<br />
ÿêà äîð³âíþº íóëþ, àëå íå ìຠâ í³é åêñòðåìóìó.<br />
Òî÷êè, â ÿêèõ âèêîíàíà íåîáõ³äíà óìîâà åêñòðåìóìó<br />
(f′(x) = 0 àáî f(x) íå äèôåðåíö³éîâíà), íàçèâàþòüñÿ êðèòè÷íèìè.<br />
Ö³ òî÷êè “ï³äîçð³ë³” íà åêñòðåìóì. Ïèòàííÿ ïðî<br />
íàÿâí³ñòü åêñòðåìóìó â êðèòè÷íèõ òî÷êàõ âèð³øóºòüñÿ äîñòàòí³ìè<br />
óìîâàìè. Òî÷êè, â ÿêèõ f′(x) = 0, çâóòüñÿ ñòàö³îíàðíèìè.<br />
7.16.2. Äîñòàòí³ óìîâè åêñòðåìóìó ôóíêö³¿<br />
Òåîðåìà 7.16.2. Íåõàé äëÿ ôóíêö³¿ f(x) òî÷êà õ 0 º<br />
êðèòè÷íîþ ³ ôóíêö³ÿ f(x) äèôåðåíö³éîâíà â äåÿêîìó îêîë³<br />
õ 0 , êð³ì, ìîæëèâî, òî÷êè õ 0 , â ÿê³é âîíà íåïåðåðâíà. Òîä³<br />
ÿêùî ïðè ïåðåõîä³ ÷åðåç õ 0 çë³âà íàïðàâî f′(x) çì³íþº çíàê,<br />
òî ôóíêö³ÿ â òî÷ö³ õ 0 ìຠëîêàëüíèé åêñòðåìóì. Ïðè÷îìó<br />
ÿêùî çíàê f′(x) çì³íþºòüñÿ ç + íà –, òî f(x) â òî÷ö³ õ 0 ìàº<br />
ëîêàëüíèé ìàêñèìóì, ÿêùî ç – íà +, òî — ëîêàëüíèé ì³í³ìóì.<br />
Äîâåäåííÿ. Ðîçãëÿíåìî âèïàäîê, êîëè çíàê ïîõ³äíî¿<br />
f′(x) çì³íþºòüñÿ ç + íà –. Òîä³ íà â³äð³çêó ì³æ õ ³ õ 0 , äå<br />
õ — áóäü-ÿêà òî÷êà îêîëó õ 0 , äëÿ f(x) âèêîíàí³ óìîâè òåîðåìè<br />
Ëàãðàíæà. Òîìó f(x) –f(x 0 )=f′(ñ) (x – x 0 ), äå ñ∈(õ, õ 0 ).<br />
Îñê³ëüêè f′(c) > 0 ïðè x < x 0 ³ f′(c) < 0 ïðè x > x 0 , òî çàâæäè<br />
f(x) –f(x 0 ) < 0, òîáòî f(x) < f(x 0 ). Öå îçíà÷àº, ùî â òî÷ö³ õ 0<br />
ôóíêö³ÿ f(x) ìຠëîêàëüíèé ìàêñèìóì.<br />
Àíàëîã³÷íî ðîçãëÿäàºòüñÿ âèïàäîê ëîêàëüíîãî ì³í³ìóìó.<br />
Äîâåäåíà äîñòàòíÿ óìîâà äຠïåðøèé ñïîñ³á äîñë³äæåííÿ<br />
ôóíêö³¿ íà åêñòðåìóì.<br />
Ïðèêëàä 7.16.1. Äîñë³äèòè íà åêñòðåìóì ôóíêö³þ<br />
2<br />
x + 2x+<br />
1<br />
y= f( x)<br />
= .<br />
x −1<br />
Ð î ç â ’ ÿ ç à í í ÿ. Ôóíêö³ÿ âèçíà÷åíà ³ äèôåðåíö³éîâíà íà<br />
ìíîæèí³ D =(–∞, 1) ∪ (1, +∞). Íà ö³é ìíîæèí³<br />
y′ =<br />
( x+ 1)( x−3)<br />
( x − 1) 2<br />
.<br />
252 253
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