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íèêó, à äðóãà òî÷êà ëåæèòü çà ìåæàìè òðèêóòíèêà<br />
(ðèñ. 10.17).<br />
Ðèñ. 10.17<br />
Òàêèì ÷èíîì äîñë³äæåííÿ íà åêñòðåìóì òðåáà ïðîâîäèòè<br />
â òî÷ö³ Î(0,0). Ìàºìî:<br />
( )<br />
A= z′′ 0,0 =− 6 < 0,<br />
xx<br />
( )<br />
C = z′′ 0,0 = − 2 ,<br />
yy<br />
2<br />
∆= AC − B = > .<br />
12 0<br />
Çã³äíî ç òåîðåìîþ 10.6.2 â òî÷ö³ Î(0,0) ôóíêö³ÿ ìàº<br />
ìàêñèìóì: zmax z( 0,0)<br />
= = 0 . Çàóâàæèìî, ùî äëÿ âêàçàíîãî<br />
ïðàâèëà íå îáîâ’ÿçêîâî çíàòè, ÷è òî÷êà Î(0,0) º òî÷êîþ<br />
ìàêñèìóìó.<br />
Äîñë³äèìî òåïåð ôóíêö³þ íà ìåæ³ òðèêóòíèêà ç âåðøèíàìè<br />
M 2 , M 3 , M 4 .<br />
Òðèêóòíèê ñêëàäàºòüñÿ ç òðüîõ ñòîð³í. Äîñë³äæåííÿ áóäåìî<br />
ïðîâîäèòè íà êîæí³é ç öèõ ñòîð³í. Íà ñòîðîí³<br />
MM<br />
4 2 ( x=−1, −2 ≤y≤ 4)<br />
äàíà ôóíêö³ÿ ïåðåòâîðþºòüñÿ ó<br />
2<br />
ôóíêö³þ îäí³º¿ çì³ííî¿ z =ψ ( y) =−4<br />
− y , −2≤y<br />
≤ 4. Íà ñåãìåíò³<br />
[–2, 4] äîñë³äæåííÿ ôóíêö³¿ ψ(y) äóæå ïðîñòå (÷èòà-<br />
÷åâ³ ðåêîìåíäóºìî öå çðîáèòè). Ðåçóëüòàòè éîãî òàê³: ïðè<br />
y = 0 ôóíêö³ÿ ψ(y) ìຠìàêñèìóì, ð³âíèé –4. Öåé ìàêñèìóì<br />
º íàéá³ëüøèì çíà÷åííÿì ôóíêö³¿ ψ(y) íà ñåãìåíò³ [–2, 4].<br />
 òî÷ö³ y = 4 çíà÷åííÿ ôóíêö³¿ ì³í³ìàëüíå ³ äîð³âíþº –20.<br />
Íà ñòîðîí³ M 4 M 3 (y =4,–1≤ x ≤ 5) äàíà ôóíêö³ÿ ïåðåòâîðþºòüñÿ<br />
ó ôóíêö³þ ϕ ( x) = x 3 −3x 2 −16, x∈[ − 1,5]<br />
. Öÿ ôóíêö³ÿ<br />
íà ñåãìåíò³ [–1,5] ìຠì³í³ìóì â òî÷ö³ x = 2 ³ äîð³âíþº –20,<br />
à ìàêñèìóì â òî÷ö³ x = 0, ÿêèé äîð³âíþº –16. Íà ìåæ³<br />
ñåãìåíòà [–1,5] ¿¿ çíà÷åííÿ â³äïîâ³äíî äîð³âíþþòü –20 ³ 34.<br />
Íàðåøò³, íà ñòîðîí³ M 3 M 2 äàíà ôóíêö³ÿ ïåðåòâîðþºòüñÿ<br />
3 2<br />
ó ôóíêö³þ ω ( x) = x − 4x + 2x− 1 , x ∈− [ 1, 5]<br />
. Ïðîñòèé àíàë³ç<br />
ïîêàçóº (ïåðåâ³ðòå!), ùî íà ñåãìåíò³ [–1, 5] ôóíêö³ÿ ω(x)<br />
äîñÿãຠíàéìåíøîãî ³ íàéá³ëüøîãî çíà÷åíü ó ìåæîâèõ òî÷êàõ,<br />
à ñàìå: ω(–1) = –8, ω(5) = 34.<br />
²ç ìíîæèíè âèä³ëåíèõ ÷èñåë 0, –4, –8, –20, –16, 34 âèáèðàºìî<br />
íàéá³ëüøå ³ íàéìåíøå. Âîíè ³ áóäóòü â³äïîâ³äàòè íàéá³ëüøîìó<br />
³ íàéìåíøîìó çíà÷åííþ ôóíêö³¿ z = x 3 –3x 2 – y 2 â<br />
òðèêóòíèêó M 2 M 3 M 4 . ßñíî, ùî íàéá³ëüøå çíà÷åííÿ äîð³âíþº<br />
34, à íàéìåíøå –20.<br />
³äçíà÷èìî òàêîæ, ùî ôóíêö³ÿ z = x 3 –3x 2 – y 2 ïðèéìàº<br />
íàéá³ëüøå çíà÷åííÿ ó âåðøèí³ M 3 , à íàéìåíøå ó âåðøèí³ M 4 .<br />
Çàóâàæåííÿ. Íàâåäåíèé ïðèêëàä º ïîâ÷àëüíèì. Ùîá öå<br />
ïîêàçàòè ïðîâåäåìî íåñêëàäíèé àíàë³ç. Ó ðîçãëÿíóòîìó ïðèêëàä³<br />
ôóíêö³ÿ ìຠò³ëüêè îäèí åêñòðåìóì, ³ â³í âèÿâèâñÿ<br />
ìàêñèìóìîì ôóíêö³¿. Òåïåð ïîð³âíÿºìî ç àíàëîã³÷íîþ ñèòóàö³ºþ<br />
äëÿ ôóíêö³¿ îäí³º¿ çì³ííî¿. Óÿâèìî ñîá³, ùî ôóíêö³ÿ<br />
îäí³º¿ çì³ííî¿ íà ñåãìåíò³ ìàëà á îäèí ìàêñèìóì, à ì³í³ìóì³â<br />
íå áóëî á. Òîä³ ìè çðîáèëè áè âèñíîâîê, ùî íàéá³ëüøå çíà÷åííÿ<br />
ôóíêö³¿ îäí³º¿ çì³ííî¿ äîð³âíþâàëî áè çíàéäåíîìó ìàêñèìóìó<br />
ö³º¿ ôóíêö³¿. Ó íàøîìó ïðèêëàä³ ôóíêö³ÿ z = x 3 –3x 2 –<br />
y 2 ìຠò³ëüêè îäíó òî÷êó ìàêñèìóìó â îáëàñò³ ³ íå ìຠòî÷îê<br />
ì³í³ìóìó. Ïðîòå íàéá³ëüøå çíà÷åííÿ íå ñï³âïàäຠç ìàêñèìóìîì,<br />
à äîð³âíþº çíà÷åííþ ôóíêö³¿ ó ìåæîâ³é òî÷ö³. Òàêèì<br />
÷èíîì, íå çàâæäè ìîæëèâèé ïåðåíîñ ôàêò³â ç ôóíêö³é îäí³º¿<br />
çì³ííî¿ äî ôóíêö³é áàãàòüîõ çì³ííèõ. Îáåðåæí³ñòü ïðè ïåðåíîñ³<br />
ôàêò³â íå çàâàäèòü ³, á³ëüø òîãî, º âåëüìè êîðèñíîþ. Ïðî<br />
òàê³ ôàêòè ìè âæå ãîâîðèëè (äèâ. ï. 10.3).<br />
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