You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Òîä³: 1) ÿêùî<br />
a a a<br />
a > 0, ∆ = > 0, a a a > 0,<br />
11 12 13<br />
a11 a12<br />
11 1 21 22 23<br />
a21 a22<br />
a31 a32 a33<br />
òî â òî÷ö³ M 0 (x 0 , y 0 , z 0 ) ôóíêö³ÿ u = f(x, y, z) ìຠì³í³ìóì;<br />
2) ÿêùî<br />
a a a<br />
a < 0, ∆ = > 0, a a a < 0,<br />
11 12 13<br />
a11 a12<br />
11 1 21 22 23<br />
a21 a22<br />
a31 a32 a33<br />
òî â òî÷ö³ M 0 (x 0 , y 0 , z 0 ) ôóíêö³ÿ u = f(x, y, z) ìຠìàêñèìóì.<br />
Ïðèêëàä 10.6.4. Çíàéòè åêñòðåìóì ôóíêö³¿:<br />
u = x 2 + y 2 + z 2 +2x +4y –6z.<br />
Ð î ç â ’ ÿ ç à í í ÿ. Ó â³äïîâ³äíîñò³ äî çàóâàæåííÿ 3<br />
ï. 10.6.1. äëÿ çíàõîäæåííÿ ñòàö³îíàðíèõ òî÷îê îòðèìàºìî<br />
òàêó ñèñòåìó<br />
⎧ u′ x<br />
= 2x<br />
+ 2 = 0<br />
⎪<br />
⎨uy<br />
′ = 2y<br />
+ 4 = 0<br />
⎪<br />
⎩uz<br />
′ = 2z<br />
− 6 = 0.<br />
Ðîçâ’ÿçàâøè ¿¿ (áóäü-ÿêèì ñïîñîáîì), çíàéäåìî ºäèíó<br />
ñòàö³îíàðíó òî÷êó M 0 (–1, –2, 3).<br />
Äàë³ çàñòîñóºìî òåîðåìó 10.6.3. Ó òî÷ö³ M 0 (–1, –2, 3)<br />
ìàòèìåìî<br />
2 0 0<br />
2 0<br />
a<br />
11<br />
= 2> 0, ∆<br />
1<br />
= = 4> 0, ∆<br />
3<br />
= 0 2 0 = 8><br />
0.<br />
0 2<br />
0 0 2<br />
Îòæå, çà çãàäàíîþ òåîðåìîþ ôóíêö³ÿ<br />
u = x 2 + y 2 + z 2 +2x +4y –6z â òî÷ö³ M 0 (–1, –2, 3) ìຠì³í³ìóì<br />
2 2 2<br />
( ) ( ) ( ) ( ) ( )<br />
u<br />
min<br />
= − 1 + − 2 + 3 + 2⋅ − 1 + 4⋅ −2 −6⋅ 3= − 14.<br />
10.6.2. Íàéá³ëüø³ òà íàéìåíø³ çíà÷åííÿ ôóíêö³¿<br />
Íåõàé ôóíêö³ÿ z = f(x, y) íåïåðåðâíà â çàìêíåí³é îáìåæåí³é<br />
îáëàñò³ D . Òîä³ çà òåîðåìîþ Âåéåðøòðàññà (òâåðäæåííÿ<br />
2) òåîðåìè 10.2.2.) â òàê³é îáëàñò³ ôóíêö³ÿ ìàº<br />
íàéìåíøå òà íàéá³ëüøå çíà÷åííÿ, òîáòî ³ñíóþòü òàê³ òî÷êè<br />
M 1 (x 1 , y 1 ) ³ M 2 (x 2 , y 2 ), ùî â îáëàñò³ íåïåðåðâíîñò³ ôóíêö³¿<br />
z = f(x, y) âèêîíóþòüñÿ óìîâè<br />
( , ) ( , ) ( , ) ( , )<br />
f x y ≤ f x y ≤ f x y ∀M x y ∈ D.<br />
1 1 2 2<br />
Òåîðåìà ãàðàíòóº ³ñíóâàííÿ òî÷îê îáëàñò³ D , â ÿê³é<br />
ôóíêö³ÿ z = f(x, y) äîñÿãຠñâîãî íàéá³ëüøîãî òà íàéìåíøîãî<br />
çíà÷åíü, àëå âîíà í³÷îãî íå ãîâîðèòü ïðî òå, ÿê ¿õ çíàéòè.<br />
Ó çàãàëüíîìó âèïàäêó òèì ïà÷å íåìຠïðàâèëà äëÿ â³äøóêàííÿ<br />
âêàçàíèõ âèùå òî÷îê.<br />
Îäíàê äëÿ ïåâíèõ êëàñ³â ôóíêö³é, ÿê³ ÷àñòî çóñòð³÷àþòüñÿ<br />
íà ïðàêòèö³, çîêðåìà â ïèòàííÿõ åêîíîì³êè, òàêå ïðàâèëî<br />
º. Ìè çàðàç ïîçíàéîìèìîñÿ ç íèì. Öå ïðàâèëî êîíñòðóêòèâíå<br />
³ áàçóºòüñÿ íà òàêîìó àëãîðèòì³:<br />
1. Çíàéòè êðèòè÷í³ òî÷êè, ÿê³ ëåæàòü óñåðåäèí³ îáëàñò³<br />
D, ³ îá÷èñëèòè çíà÷åííÿ ôóíêö³¿ â öèõ òî÷êàõ (íå âäàþ÷èñü<br />
â äîñë³äæåííÿ, ÷è áóäå â íèõ åêñòðåìóì ôóíêö³¿ ³ ÿêîãî<br />
âèäó).<br />
2. Çíàéòè íàéá³ëüøå (íàéìåíøå) çíà÷åííÿ ôóíêö³¿ íà<br />
ìåæ³ îáëàñò³ D.<br />
3. Ïîð³âíÿòè îòðèìàí³ çíà÷åííÿ ôóíêö³¿: ñàìå á³ëüøå<br />
(ìåíøå) ç íèõ ³ áóäå íàéá³ëüøèì (íàéìåíøèì) çíà÷åííÿì<br />
ôóíêö³¿ ó âñ³é îáëàñò³ D .<br />
Ïðèêëàä 10.6.5. Â òðèêóòíèêó, ÿêèé îáìåæåíèé ïðÿìèìè<br />
x = –1, y =4 ³ y = x – 1, çíàéòè íàéá³ëüøå òà íàéìåíøå<br />
çíà÷åííÿ ôóíêö³¿ z = x 3 –3x 2 – y 2 .<br />
Ð î ç â ’ ÿ ç à í í ÿ. ²ç ñèñòåìè<br />
⎧′= ⎪zx<br />
x − x=<br />
⎨<br />
⎪⎩ zy<br />
′ =− 2y<br />
= 0<br />
2<br />
3 6 0<br />
çíàõîäèìî äâ³ êðèòè÷í³ òî÷êè M 0 (0, 0) ³ M 1 (2, 0). Ïåðøà<br />
òî÷êà ñï³âïàäຠç ïî÷àòêîì êîîðäèíàò ³ íàëåæèòü òðèêóò-<br />
384 385
Change language