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Íàñë³äîê 3. ßêùî íà ³íòåðâàë³ (a, b) äèôåðåíö³éîâíà<br />
ôóíêö³ÿ f(x) òàêà, ùî f′(x) >0 (f′(x) < 0), òî íà öüîìó ³íòåðâàë³<br />
ôóíêö³ÿ f(x) çðîñòຠ(ñïàäàº).<br />
Äîâåäåííÿ. Íå îáìåæóþ÷è çàãàëüíîñò³, ïðèïóñòèìî,<br />
ùî f′(x) >0∀x ∈(a, b). Îñê³ëüêè çà óìîâîþ ôóíêö³ÿ f(x)<br />
äèôåðåíö³éîâíà íà ³íòåðâàë³ (a, b), òî äëÿ áóäü-ÿêèõ x 1 i<br />
x 2 (x 1 < x 2 ) çã³äíî ç òåîðåìîþ Ëàãðàíæà ìàºìî<br />
fx ( ) − fx ( ) = f′<br />
( ξ)( x − x),<br />
x < ξ < x. (7.13.8)<br />
2 1 2 1 1 2<br />
Î÷åâèäíî, ùî ïðàâà ÷àñòèíà ð³âíîñò³ (7.13.8) á³ëüøå<br />
íóëÿ. Öå â ñâîþ ÷åðãó ïðèâîäèòü äî òîãî, ùî<br />
fx ( ) < fx ( ) ∀x ∈( ab , ) i ∀x ∈ ( ab , ), äå x < x.<br />
1 2 1 2 1 2<br />
Îñòàíí³ äâ³ íåð³âíîñò³ ³ ïîêàçóþòü, ùî ôóíêö³ÿ f(x) ä³éñíî<br />
çðîñòຠíà ³íòåðâàë³ (a, b).<br />
Àíàëîã³÷íî äîâîäèòüñÿ äðóãà ÷àñòèíà òåîðåìè (âèïàäîê<br />
f′(x) < 0). Ïðîïîíóºìî öå çä³éñíèòè ÷èòà÷åâ³.<br />
Òåîðåìà 7.13.4 (Êîø³). ßêùî ôóíêö³¿ f(x) ³ g(x) íåïåðåðâí³<br />
íà ñåãìåíò³ [a, b], äèôåðåíö³éîâí³ íà ³íòåðâàë³ (a, b)<br />
³ g′(x) ≠ 0 ∀x∈ ( a, b)<br />
, òî çíàéäåòüñÿ òàêà òî÷êà ξ∈(a, b), â ÿê³é<br />
âèêîíóþòüñÿ ñï³ââ³äíîøåííÿ<br />
( ) − ( )<br />
( ) ( )<br />
( ξ)<br />
( )<br />
f b f a f′<br />
=<br />
g b − g a g ′ ξ<br />
. (7.13.9)<br />
Ä î â å ä å í í ÿ. Ââåäåìî äîïîì³æíó ôóíêö³þ (x), ÿêà çàäîâîëüíÿº<br />
âèìîãè òåîðåìè Ðîëëÿ<br />
( ) − f( a)<br />
( ( ) ( ))<br />
( ) − g( a)<br />
f b<br />
( x) = f( x) −f( a)<br />
− g x −g a<br />
g b<br />
³äçíà÷èìî, ùî g(a) ≠ g(b). Öå âèïëèâຠç óìîâ äàíî¿ òåîðåìè<br />
³ òåîðåìè Ðîëëÿ.<br />
Òîä³ ∃ ξ∈(a, b), ùî ′(ξ) = 0, òîáòî<br />
( ) ( )<br />
( ) ( )<br />
( ) ( )<br />
( ) ( )<br />
.<br />
( )<br />
( )<br />
f b −f a f b −f a f′<br />
ξ<br />
′ ( ξ ) = f′ ( ξ)<br />
− g′<br />
( ξ ) = 0 ⇒ =<br />
g b − g a g b − g a g ′ ξ<br />
.<br />
7.14. ÐÎÇÊÐÈÒÒß ÍÅÂÈÇÍÀ×ÅÍÎÑÒÅÉ<br />
7.14.1. Ïðàâèëî Ëîï³òàëÿ 1<br />
Åôåêòèâíèì çàñîáîì äëÿ çíàõîäæåííÿ ãðàíèö³ ôóíêö³¿<br />
â îñîáëèâèõ âèïàäêàõ º òàêå ïðàâèëî Ëîï³òàëÿ: ãðàíèöÿ<br />
â³äíîøåííÿ äâîõ íåñê³í÷åííî ìàëèõ àáî äâîõ íåñê³í÷åííî<br />
âåëèêèõ ôóíêö³é äîð³âíþº ãðàíèö³ â³äíîøåííÿ ¿õ ïîõ³äíèõ<br />
(ÿêùî îñòàííÿ ãðàíèöÿ ³ñíóº àáî äîð³âíþº íåñê³í÷åííîñò³).<br />
Öå ïðàâèëî ïîäàìî áåç äîâåäåííÿ. ßêùî ÷èòà÷à çàö³êàâèòü<br />
äîâåäåííÿ ïðàâèëà Ëîï³òàëÿ äëÿ ðîçêðèòòÿ íåâèçíà÷åíîñòåé<br />
0/0 (äèâ. ï. 6.22), òî â³í öå ìîæå çä³éñíèòè çà äîïîìîãîþ<br />
òåîðåìè Êîø³. Äîâåäåííÿ ïðàâèëà Ëîï³òàëÿ äëÿ<br />
ðîçêðèòòÿ íåâèçíà÷åíîñòåé ∞/∞ çä³éñíþºòüñÿ çíà÷íî ñêëàäí³øå.<br />
7.14.2. Çàñòîñóâàííÿ ïðàâèëà Ëîï³òàëÿ äëÿ ðîçêðèòòÿ<br />
íåâèçíà÷åíîñòåé 0/0 òà ∞/∞<br />
Íåâèçíà÷åí³ñòü 0/0 ÿâëÿº ñîáîþ â³äíîøåííÿ äâîõ íåñê³í-<br />
÷åííî ìàëèõ ôóíêö³é, à íåâèçíà÷åí³ñòü ∞/∞ ÿâëÿº ñîáîþ<br />
â³äíîøåííÿ äâîõ íåñê³í÷åííî âåëèêèõ ôóíêö³é. Çã³äíî ç<br />
ïðàâèëîì Ëîï³òàëÿ, ÿêùî f 1 (x) ³ f 2 (x) îäíî÷àñíî ïðÿìóþòü<br />
äî íóëÿ àáî äî íåñê³í÷åííîñò³ ïðè x → a àáî x →∞, òî<br />
( )<br />
( )<br />
2 2<br />
( )<br />
( )<br />
f1 x f′<br />
1<br />
x<br />
lim = lim<br />
f x f′ x<br />
,<br />
çà óìîâè, ùî ³ñíóº ñê³í÷åííà àáî íåñê³í÷åííà ãðàíèöÿ ïðàâî¿<br />
÷àñòèíè. ßêùî â³äíîøåííÿ ïîõ³äíèõ òàêîæ áóäå ÿâëÿòè<br />
âèïàäîê 0/0 àáî ∞/∞, òî ìîæíà çíîâó ³ çíîâó çàñòîñîâóâàòè<br />
ïðàâèëî Ëîï³òàëÿ, ÿêùî öå äîö³ëüíî ³ äຠðåçóëüòàò.<br />
Ïðèêëàäè 7.14.1 — 7.14.7. Çíàéòè ãðàíèö³:<br />
7.14.1.<br />
4<br />
m m<br />
x −16<br />
x − a<br />
lim<br />
x→2<br />
3 2 ; 7.14.2. lim , a ≠ 0<br />
x + 5x −6x−16<br />
x→a<br />
n n ;<br />
x − a<br />
1<br />
Ëîï³òàëü Ôðàíñóà (1661 – 1704) — ôðàíöóçüêèé ìàòåìàòèê.<br />
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