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Òåîðåìà 7.13.2 (Ðîëëÿ 1 ïðî êîð³íü ïîõ³äíî¿). ßêùî<br />
ôóíêö³ÿ f(õ) íåïåðåðâíà íà ñåãìåíò³ [a, b], äèôåðåíö³éîâíà<br />
íà ³íòåðâàë³ (à, b) ³ f(a) =f(b), òî ³ñíóº õî÷à á îäíà òàêà<br />
òî÷êà ξ∈(à, b), ùî f′(ξ) =0.<br />
Ä î â å ä å í í ÿ. 1) ÿêùî f(õ) = const, òîáòî f(x) =f(a) =f(b),<br />
x∈(a, b), òî f′(ξ) = 0 äëÿ ∀ξ∈(a, b); 2) f(x) ≠ const. Îñê³ëüêè<br />
f(a) =f(b) ³ f(x) íåïåðåðâíà, òî íà ³íòåðâàë³ (a, b) âîíà ïîâèííà<br />
äîñÿãàòè, ïðèíàéìí³, ñâîãî íàéá³ëüøîãî àáî íàéìåíøîãî<br />
çíà÷åííÿ (ìîæëèâî ³ òå ³ äðóãå, ÿêùî òàêèìè íå<br />
º f(a) =f(b)). Îòæå, çíàéäåòüñÿ õî÷à á îäíà òî÷êà ξ∈(à, b) ëîêàëüíîãî<br />
åêñòðåìóìó. Àëå òîä³ ó ö³é òî÷ö³ çà òåîðåìîþ<br />
Ôåðìà f′(ξ) =0.<br />
Òåîðåìà 7.13.3 (Ëàãðàíæà 2 ïðî ñê³í÷åííèé ïðèð³ñò<br />
ôóíêö³¿). ßêùî ôóíêö³ÿ f(x) íåïåðåðâíà íà ñåãìåíò³ [a, b],<br />
äèôåðåíö³éîâíà íà ³íòåðâàë³ (a, b), òî çíàéäåòüñÿ òàêà òî÷êà<br />
ξ∈(a, b), ùî<br />
( ) − f( a)<br />
f b<br />
b − a<br />
( )<br />
= f′<br />
ξ . (7.13.2)<br />
Ä î â å ä å í í ÿ. Ïîáóäóºìî äîïîì³æíó ôóíêö³þ (õ), ÿêà<br />
çàäîâîëüíÿëà á òåîðåìó Ðîëëÿ:<br />
( ) − f( a) ( ) , ( ( ) ( ) 0 )<br />
f b<br />
( x) = f( x) −f( a)<br />
− x− a a = b =<br />
b−<br />
a<br />
Çà òåîðåìîþ Ðîëëÿ ³ñíóº òàêà òî÷êà ξ∈(a, b), ùî ′(ξ) =0.<br />
Òîä³<br />
( ) − ( ) ( ) − ( )<br />
f b f a f b f a<br />
′ ( ξ ) = f′ ( ξ)<br />
− = 0 ⇒ = f′<br />
( ξ)<br />
.<br />
b −a b −a<br />
Î÷åâèäíî, ùî òåîðåìà Ðîëëÿ º îêðåìèì âèïàäêîì òåîðåìè<br />
Ëàãðàíæà (ïðè f(a) =f(b)).<br />
Çàóâàæèìî òàêîæ, ùî ³ç ð³âíîñò³ (7.13.2) âèïëèâຠòàê<br />
íàçâàíà ôîðìóëà Ëàãðàíæà:<br />
f(b) –f(a) =f′(ξ)(b – a). (7.13.3)<br />
1<br />
Ðîëëü ̳øåëü (1652 – 1719) — ôðàíöóçüêèé ìàòåìàòèê.<br />
2<br />
Ëàãðàíæ Ëó³ (1736 – 1813) — âèäàòíèé ôðàíöóçüêèé ìàòåìàòèê ³<br />
ìåõàí³ê.<br />
.<br />
Ç òåîðåìè Ëàãðàíæà âèïëèâàþòü òàê³ íàñë³äêè:<br />
Íàñë³äîê 1 (êðèòåð³é ñòàëîñò³ äèôåðåíö³éîâíî¿ ôóíêö³¿<br />
íà ³íòåðâàë³). Äëÿ òîãî ùîá äèôåðåíö³éîâíà íà ³íòåðâàë³<br />
(a, b) ôóíêö³ÿ f(x) äîð³âíþâàëà ñòàë³é, íåîáõ³äíî ³ äîñòàòíüî,<br />
ùîá f′(x) =0∀x ∈(a, b).<br />
Íåîáõ³äí³ñòü. Íåõàé f(x) =C ∀x ∈(a, b), äå C — ñòàëà.<br />
ßñíî, ùî f′(x) =0∀x ∈(a, b).<br />
Äîñòàòí³ñòü. Íåõàé f′(x) =0∀x ∈(a, b). ³çüìåìî íà<br />
³íòåðâàë³ (a,b) äâ³ äîâ³ëüí³ òî÷êè x 1 i x 2 (x 1 < x 2 ). Òîä³ ôóíêö³ÿ<br />
f(x) íà ñåãìåíò³ [x 1 , x 2 ] çàäîâîëüíÿº âñ³ óìîâè òåîðåìè<br />
Ëàãðàíæà. Âíàñë³äîê ÷îãî âèêîíóºòüñÿ ð³âí³ñòü<br />
fx ( ) − fx ( ) = f′<br />
( ξ)( x − x),<br />
x < ξ < x. (7.13.4)<br />
2 1 2 1 1 2<br />
Çà óìîâîþ f′(x) =0∀x ∈(a, b), çîêðåìà é ïðè x = ξ, òîáòî<br />
f′(ξ) = 0. Òîä³ ç ð³âíîñò³ (7.13.4) âèïëèâàº, ùî<br />
f(x 2 )–f(x 1 ) = 0, àáî<br />
f(x 2 )=f(x 1 ). (7.13.5)<br />
Îñê³ëüêè òî÷êè x 1 i x 2 äîâ³ëüí³, òî ð³âí³ñòü (7.13.5) îçíà-<br />
÷àº, ùî f(x) =C ∀x ∈(a, b), äå C — ñòàëà, ùî ³ òðåáà áóëî<br />
äîâåñòè.<br />
Íàñë³äîê 2 (îñíîâíà ëåìà ³íòåãðàëüíîãî ÷èñëåííÿ).<br />
Äëÿ òîãî ùîá äâ³ äèôåðåíö³éîâí³ íà ³íòåðâàë³ (a, b) ôóíêö³¿<br />
f(x) i ϕ(x) â³äð³çíÿëèñÿ íà ñòàëó, íåîáõ³äíî ³ äîñòàòíüî,<br />
ùîá ¿õí³ ïîõ³äí³ ñï³âïàäàëè, òîáòî<br />
f′(x) =ϕ′(x) ∀x ∈(a, b). (7.13.6)<br />
Í å î á õ³ ä í ³ ñ ò ü. Íåõàé<br />
f(x) –ϕ(x) =C ∀x ∈(a, b). (7.13.7)<br />
Î÷åâèäíî, ùî ïðè öüîìó âèêîíóºòüñÿ óìîâà (7.13.6).<br />
Äîñòàòí³ñòü. Íåõàé òåïåð ìຠì³ñöå (7.13.6). Ïîçíà÷èìî<br />
ð³çíèöþ f(x) –ϕ(x) ÷åðåç ψ(x): ψ(x) =f(x) –ϕ(x). Òîä³<br />
ôóíêö³ÿ ψ(x) ìຠïîõ³äíó ³ ψ′(x) =f′(x) –ϕ′(x) =0.<br />
Çâ³äñè çã³äíî ç íàñë³äêîì 1 âèïëèâàº, ùî ψ(x) =C, àáî<br />
f(x) –ϕ(x) =C ∀x ∈(a, b), ùî é òðåáà áóëî äîâåñòè.<br />
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