You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Îòæå, ñòàö³îíàðíà òî÷êà ò³ëüêè îäíà — öå ïî÷àòîê êîîðäèíàò<br />
Î(0,0). Àëå öÿ òî÷êà íå º òî÷êîþ åêñòðåìóìó ôóíêö³¿<br />
z = x 2 – y 2 , îñê³ëüêè ó â³äïîâ³äíîñò³ äî ¿¿ ñòðóêòóðè âîíà â<br />
áóäü-ÿêîìó ìàëîìó îêîë³ òî÷êè Î(0,0) ïðèéìຠÿê äîäàòí³,<br />
òàê ³ â³ä’ºìí³ çíà÷åííÿ.<br />
2 2<br />
Ïðèêëàä 10.6.2. z = x + y .<br />
Ãðàô³ê ö³º¿ ôóíêö³¿ º êîíóñ. Ïîâåðõíþ, ÿêà â³äïîâ³äàº<br />
2 2<br />
z = x + y , ìîæíà ïîáóäóâàòè øëÿõîì îáåðòàííÿ<br />
ôóíêö³¿<br />
ãðàô³êà ôóíêö³¿ z<br />
= x íàâêîëî â³ñ³ Oz.<br />
2 2<br />
 òî÷ö³ Î(0,0) ôóíêö³ÿ z = x + y ìຠÿâíèé ì³í³ìóì,<br />
íåçâàæàþ÷è íà òå, ùî âîíà º ò³ëüêè òî÷êîþ “ï³äîçð³ëîþ íà<br />
åêñòðåìóì” (÷àñòèíí³ ïîõ³äí³ ðîçãëÿäóâàíî¿ ôóíêö³¿<br />
z′ =<br />
x<br />
x<br />
x<br />
+ y<br />
2 2<br />
³<br />
z′ =<br />
y<br />
x<br />
y<br />
+ y<br />
2 2<br />
â òî÷ö³ Î(0,0) íå ³ñíóþòü).<br />
Îòæå, ðîçãëÿíóò³ êîíòðïðèêëàäè ïîêàçóþòü, ùî ä³éñíî, íå<br />
âñ³ êðèòè÷í³ òî÷êè ìîæóòü áóòè òî÷êàìè åêñòðåìóìó ³<br />
òîìó âèíèêຠïðîáëåìà ïîøóêó äîñòàòí³õ óìîâ ³ñíóâàííÿ<br />
åêñòðåìóìó ôóíêö³¿ äâîõ çì³ííèõ â êðèòè÷íèõ òî÷êàõ.<br />
Äîñòàòí³ óìîâè ó âèïàäêó ñòàö³îíàðíèõ òî÷îê áóëî ìàòåìàòèêàìè<br />
çíàéäåíî.<br />
Ò å î ð å ì à 10.6.2. (ïðî äîñòàòíþ óìîâó ³ñíóâàííÿ åêñòðåìóìó<br />
ôóíêö³¿ äâîõ çì³ííèõ). Íåõàé ôóíêö³ÿ z = f(x, y)<br />
âèçíà÷åíà â δ-îêîë³ ñòàö³îíàðíî¿ òî÷êè M 0 (x 0 , y 0 ). Êð³ì<br />
öüîãî, â í³é äàíà ôóíêö³ÿ ìຠíåïåðåðâí³ ÷àñòèíí³ ïîõ³äí³<br />
äðóãîãî ïîðÿäêó<br />
( , ) , ( , ) ( , ) , ( , )<br />
z′′ x y = A z′′ x y = z′′ x y = B z′′<br />
x y = C.<br />
xx 0 0 xy 0 0 yx 0 0 yy 0 0<br />
Òîä³, 1) ÿêùî ∆ = AC – B 2 > 0, òî â òî÷ö³ M 0 (x 0 , y 0 ) ôóíêö³ÿ<br />
z = f(x, y) ìຠåêñòðåìóì, ïðè÷îìó, êîëè A > 0, — ì³í³ìóì,<br />
à êîëè A < 0, — ìàêñèìóì; 2) ÿêùî ∆ = AC – B 2 <br />
0,<br />
−2 4<br />
A= z′′<br />
xx<br />
4,2 = 2 > 0 . Òàêèì ÷èíîì, ó òî÷ö³ M 0 (4,2) ôóíêö³ÿ<br />
ìຠì³í³ìóì:<br />
( )<br />
2 2<br />
zmin = z 4,2 = 4 −2 ⋅4 ⋅ 2 + 2 ⋅2 −4 ⋅ 4 = − 8 .<br />
Òåîðåìà 10.6.3 (ïðî äîñòàòíþ óìîâó ³ñíóâàííÿ åêñòðåìóìó<br />
ôóíêö³¿ òðüîõ çì³ííèõ). Íåõàé ôóíêö³ÿ u = f(x, y, z)<br />
âèçíà÷åíà â δ-îêîë³ ñòàö³îíàðíî¿ òî÷êè M 0 (x 0 , y 0 , z 0 ). Êð³ì<br />
öüîãî, â í³é äàíà ôóíêö³ÿ ìຠíåïåðåðâí³ ÷àñòèíí³ ïîõ³äí³<br />
( ) ( ) ( )<br />
u′′ x , y , z = a , u′′ x , y , z = u′′<br />
x , y , z = a = a ,<br />
xx 0 0 0 11 xy 0 0 0 yx 0 0 0 12 21<br />
( ) ( ) ( )<br />
u′′ x , y , z = a , u′′ x , y , z = u′′<br />
x , y , z = a = a ,<br />
yy 0 0 0 22 xz 0 0 0 zx 0 0 0 13 31<br />
( ) ( ) ( )<br />
u′′ x , y , z = a , u′′ x , y , z = u′′<br />
x , y , z = a = a ,<br />
zz 0 0 0 33 yz 0 0 0 zy 0 0 0 23 32<br />
u′′ ( x , y , z ) = u′′<br />
( x , y , z ) = a = a .<br />
yz<br />
0 0 0 zy 0 0 0 23 32<br />
382 383
Change language