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ÂÏÐÀÂÈ<br />
Äîñë³äèòè íà åêñòðåìóì ôóíêö³¿:<br />
10.20. z = x 2 – xy + y 2 +9x –6y + 20.<br />
10.21. z = x 2 – xy + y 2 .<br />
10.22. z = x 2 –2xy +2y 2 +2x.<br />
10.23. z = x 3 + y 3 – x 2 –2xy + y 3 .<br />
10.24. u =2x 2 – xy +2xz – y + y 3 + z 2 .<br />
10.25. u =2x 2 + y 2 + z 2 –2xy +4z – x.<br />
10.7. ÓÌÎÂÍÈÉ ÅÊÑÒÐÅÌÓÌ<br />
10.7.1. Îñíîâí³ ïîíÿòòÿ ³ îçíà÷åííÿ<br />
Íåõàé çàäàíà ôóíêö³ÿ u = f(x, y), ÿêà âèçíà÷åíà â îáëàñò³<br />
D (ðèñ. 10.18, à), ³ íåõàé â ö³é îáëàñò³ çàäàíà äåÿêà ë³í³ÿ<br />
L, ð³âíÿííÿ ÿêî¿ ϕ(õ, ó) = 0 (ðèñ. 10.18, á).<br />
à<br />
á<br />
Ðèñ. 10.18<br />
Ðîçãëÿäàþ÷è ïèòàííÿ ïðî åêñòðåìóì ôóíêö³¿ u = f(x, y)<br />
â îáëàñò³ D, ìîæíà ñòàâèòè äâ³ çàäà÷³: âèçíà÷èòè åêñòðåìóì<br />
ôóíêö³¿ u = f(x, y) â îáëàñò³ D ³ åêñòðåìóì ôóíêö³¿<br />
f(x, y) íà ë³í³¿ L, ÿêà íàëåæèòü ö³º¿ îáëàñò³.  ïåðøîìó<br />
âèïàäêó êàæóòü ïðî áåçóìîâíèé åêñòðåìóì, ó äðóãîìó —<br />
ïðî óìîâíèé. Îñòàííÿ íàçâà ïîâ’ÿçàíà ç òèì, ùî íà çì³íí³<br />
õ ³ ó íàêëàäåíî äîäàòêîâó óìîâó ϕ(õ, ó) = 0. ßêùî öå ð³âíÿííÿ<br />
ðîçâ’ÿçíå, íàïðèêëàä â³äíîñíî ó = ψ(õ), òî, ï³äñòàâëÿþ÷è<br />
ó = ψ(õ) äî âèðàçó äëÿ u = f(x, y), îòðèìàºìî ñêëàäåíó<br />
ôóíêö³þ îäí³º¿ çì³ííî¿ u = f(x, ψ(x)).<br />
Ôóíêö³ÿ ϕ(õ, ó) = 0, ùî çàäຠë³í³þ L, íàçèâàºòüñÿ çâ’ÿçêîì<br />
(óìîâîþ). гâíÿííÿ ë³í³¿ L ìîæå áóòè çàäàíî ïàðàìåòðè÷íî<br />
x = x(t), y = y(t).<br />
 çàãàëüíîìó âèïàäêó çàäà÷à çíàõîäæåííÿ óìîâíîãî åêñòðåìóìó<br />
ôîðìóëþºòüñÿ òàê: çíàéòè åêñòðåìóì ôóíêö³¿<br />
u = f(x 1 , x 2 ,…, x n ) íà m-âèì³ðí³é ïîâåðõí³, ÿêà çàäàíà ð³âíÿííÿìè<br />
ϕ j (x 1 , x 2 ,…, x n ) = 0, j = 1, m, m < n.<br />
Çàäà÷³ íà óìîâíèé åêñòðåìóì çâè÷àéíî çâîäÿòü äî çàäà-<br />
÷³ íà áåçóìîâíèé åêñòðåìóì. Ðîçãëÿíåìî öå íà ïðèêëàä³<br />
äèôåðåíö³éîâíî¿ ôóíêö³¿ äâîõ çì³ííèõ u = f(x, y) ³ ïîò³ì<br />
óçàãàëüíèìî íà âèïàäîê n çì³ííèõ.<br />
Íåõàé õ, ó ïîâ’ÿçàí³ ð³âíÿííÿì ϕ(õ, ó) = 0. Ðîçãëÿäàþ÷è<br />
ôóíêö³þ u = f(x, y) ³ çâ’ÿçîê ϕ(õ, ó) =0 (ϕ(õ, ó) — äèôåðåíö³éîâíà<br />
ÿê ôóíêö³ÿ äâîõ àðãóìåíò³â õ ³ ó), îá÷èñëèìî çà<br />
ôîðìóëîþ (10.4.7) ¿õ ïîâí³ äèôåðåíö³àëè. Îòðèìàºìî<br />
∂u<br />
∂u<br />
∂ϕ ∂ϕ<br />
du = dx + dy, dϕ= dx + dy = 0<br />
∂x ∂y ∂x ∂y<br />
. (10.7.1)<br />
 ñòàö³îíàðíèõ òî÷êàõ du = 0. Îòæå,<br />
∂u<br />
∂u<br />
dx + dy = 0<br />
∂x<br />
∂y<br />
∂u<br />
∂udy<br />
àáî + = 0<br />
∂x<br />
∂y dx<br />
. (10.7.2)<br />
Ïîìíîæóþ÷è äðóãå ð³âíÿííÿ (10.7.1) íà ñòàëèé ìíîæíèê<br />
λ ³ äîäàþ÷è éîãî ï³ñëÿ ìíîæåííÿ ³ ä³ëåííÿ íà dx äî<br />
ð³âíÿííÿ (10.7.2), îòðèìàºìî<br />
àáî<br />
∂u ∂u dy ⎛∂ϕ ∂ϕdy⎞<br />
+ +λ ⎜ + ⎟ = 0 ,<br />
∂x ∂y dx ⎝∂x ∂y dx⎠<br />
∂u ∂ϕ ⎛∂u ∂ϕ⎞dy<br />
+λ + ⎜ +λ ⎟ = 0 . (10.7.3)<br />
∂x ∂x ⎝∂y ∂y⎠dx<br />
Ç ð³âíÿííÿ (10.7.3) âèçíà÷àºìî ñòàö³îíàðí³ òî÷êè, îáðàâøè<br />
ïàðàìåòð λ òàê, ùîá<br />
∂u<br />
∂ϕ<br />
+λ = 0 .<br />
∂y<br />
∂y<br />
388 389
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