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Òîä³ ð³âíÿííÿ (10.7.3) ìîæíà çàïèñàòè ó âèãëÿä³:<br />
∂u<br />
∂ϕ<br />
+λ = 0 .<br />
∂x<br />
∂x<br />
Îñòàòî÷íî ñòàö³îíàðí³ òî÷êè óìîâíîãî åêñòðåìóìó âèçíà-<br />
÷àþòüñÿ ç ñèñòåìè òðüîõ ð³âíÿíü<br />
⎧ ∂f<br />
∂ϕ<br />
⎪ +λ = 0,<br />
∂ x ∂ x<br />
⎪<br />
⎪ ∂ f ∂ϕ<br />
⎨ +λ = 0,<br />
⎪ ∂ y ∂ y<br />
⎪ϕ ( xy , ) = 0.<br />
⎪<br />
⎩<br />
ßêùî ðîçãëÿíóòè ôóíêö³þ ( x, y, ) f( x, y) ( x,<br />
y)<br />
(10.7.4)<br />
λ = +λϕ , òî<br />
çíàõîäæåííÿ óìîâíîãî åêñòðåìóìó ôóíêö³¿ f(x, y) çâåäåòüñÿ<br />
äî çíàõîäæåííÿ áåçóìîâíîãî åêñòðåìóìó ôóíêö³¿ (x, y, λ),<br />
îñê³ëüêè ñèñòåìà (10.7.4) ð³âíîñèëüíà ñèñòåì³<br />
⎧∂<br />
⎪ = 0,<br />
⎪ ∂ x<br />
⎪ ∂ <br />
⎨ = 0,<br />
⎪ ∂ y<br />
⎪∂<br />
⎪ = 0.<br />
⎩ ∂λ<br />
Ôóíêö³ÿ íàçèâàºòüñÿ ôóíêö³ºþ Ëàãðàíæà. Õàðàêòåð<br />
óìîâíîãî åêñòðåìóìó, òàê ñàìî, ÿê ³ áåçóìîâíîãî, âèçíà÷à-<br />
ºòüñÿ çà òåîðåìîþ 10.6.2.<br />
ßêùî ñòàâèòüñÿ çàäà÷à íà åêñòðåìóì ôóíêö³¿<br />
u = f(x 1 , x 2 ,…, x n ) ³ç çâ’ÿçêàìè<br />
ϕ j (x 1 , x 2 ,…, x n ) = 0, j=1,m , m < n, (10.7.5)<br />
òî ñêëàäàºòüñÿ ôóíêö³ÿ Ëàãðàíæà<br />
m<br />
=f(x,x, K,x)+ ∑ λϕ(x,x, K ,x). (10.7.6)<br />
1 2 n j j 1 2 n<br />
j=<br />
1<br />
∂<br />
Íåîáõ³äí³ óìîâè = 0( i = 1, n)<br />
∂x<br />
, ðàçîì ç ð³âíÿííÿìè<br />
i<br />
(10.7.5), óòâîðþþòü ñèñòåìó ð³âíÿíü, ç ÿêî¿ âèçíà÷àþòüñÿ<br />
êîîðäèíàòè ñòàö³îíàðíèõ òî÷îê. Ôóíêö³ÿ (10.7.6) çâîäèòü<br />
çàäà÷ó óìîâíîãî åêñòðåìóìó äî áåçóìîâíîãî.<br />
Ïðèêëàä 10.7.1. ϳäïðèºìñòâî âèð³øèëî ùîì³ñÿöÿ âèä³ëÿòè<br />
140 000 ãðí íà âèðîáíèöòâî íîâî¿ ïðîäóêö³¿. Ñåðåäíÿ<br />
çàðîá³òíà ïëàòà íà ï³äïðèºìñòâ³ äîð³âíþº 400 ãðí., à âàðò³ñòü<br />
îäèíèö³ ñèðîâèíè – 100 ãðí. Ïîòð³áíî âèçíà÷èòè, ÿêó<br />
ê³ëüê³ñòü ðîáî÷èõ k ³ ÿêó ê³ëüê³ñòü ñèðîâèíè c íåîáõ³äíî<br />
ïðèäáàòè ï³äïðèºìñòâó äëÿ îäåðæàííÿ íàéá³ëüøîãî îáñÿãó<br />
ïðîäóêö³¿ Q, ÿêùî â³äîìî, ùî<br />
Q =80k +20c – k 2 – c 2 + kc. (10.7.7)<br />
Çã³äíî ç óìîâîþ âåëè÷èíè ñ ³ k ïîâ’ÿçàí³ ì³æ cîáîþ òàê:<br />
400k + 100c = 140 000. (10.7.8)<br />
Ð î ç â ’ ÿ ç à í í ÿ. Öå çàäà÷à íà óìîâíèé åêñòðåìóì<br />
2 2<br />
( , ) 80 20 , 400 100 140000<br />
Qkc<br />
14444444444444424444444444444<br />
= k+ c −k − c + kc k + c =<br />
43<br />
2 2<br />
( , , ) 80 20 (400 100 14000)<br />
Lkcλ = k+ c−k − c + kc+λ k+ c−<br />
⎧∂L<br />
⎪ = 80 − 2k+ c + 400 λ = 0<br />
∂ k<br />
⎪ ⎧ c − 2k<br />
+ 400λ = −80<br />
⎪∂<br />
L<br />
⎪<br />
⎨ = 20 − 2c + k + 100 λ = 0 ⇒ ⎨k − 2c+ 100 λ = −20<br />
⎪∂c<br />
⎪ 400k<br />
+ 100 c = 14000.<br />
⎪∂L<br />
⎩<br />
⎪ = 400k<br />
+ 100c<br />
− 140000 = 0<br />
⎩ ∂λ<br />
Ðîçâ’ÿçóþ÷è îñòàííþ ñèñòåìó áóäü-ÿêèì ñïîñîáîì, îòðèìàºìî<br />
(ïåðåâ³ðòå!):<br />
k = 30, c = 20, λ =–10 –1 .<br />
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