Математика. Жулева Л.Д - 84 стр.

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84 3. ëÏÎÔÒÏÌØÎÁÑ ÒÁÂÏÔÁ ½6
ðÅÒÅÊÄÅÍ Ë ËÁÎÏÎÉÞÅÓËÏÊ ÚÁÄÁÞÅ; ÄÌÑ ÜÔÏÇÏ ××ÅÄÅÍ ÄÏÐÏÌÎÉÔÅÌØÎÙÅ ÐÅ-
ÒÅÍÅÎÎÙÅ x
3
, x
4
, x
5
, ÔÏÇÄÁ:
x
1
+ 2x
2
+ x
3
= 220,
2x
1
+ x
2
+ x
4
= 260,
4x
1
+ 5x
2
+ x
5
= 640,
x
1
> 0, i = 1, . . . , 5,
f(x
1
, x
2
, x
3
, x
4
, x
5
) = −8x
1
− 12x
2
− 0 · x
3
− 0 ·x
4
− 0 ·x
5
.
îÁÞÁÌØÎÏÅ ÏÐÏÒÎÏÅ ÒÅÛÅÎÉÅ ×ÏÚØÍÅÍ × ×ÉÄÅ:
x
(1)
1
= x
(1)
2
= 0, x
(1)
3
= 220, x
(1)
4
= 260, x
(1)
5
= 640,
ÔÏÇÄÁ f = 0.
ðÅÒÅÍÅÎÎÙÅ x
3
, x
4
, x
5
¡ ÂÁÚÉÓÎÙÅ, Á x
1
, x
2
¡ Ó×ÏÂÏÄÎÙÅ:
x
3
= 220 −x
1
− 2x
2
,
x
4
= 260 −2x
1
− x
2
, (∗)
x
5
= 640 −4x
1
− 5x
2
.
÷ ÆÏÒÍÕ f É x
1
É x
2
×ÈÏÄÑÔ Ó ÏÔÒÉÃÁÔÅÌØÎÙÍÉ ËÏÜÆÆÉÃÉÅÎÔÁÍÉ. âÕÄÅÍ
ÉÚÍÅÎÑÔØ x
1
, ÎÁÉÂÏÌØÛÅÅ ×ÏÚÍÏÖÎÏÅ ÚÎÁÞÅÎÉÅ, ÐÒÉ ËÏÔÏÒÏÍ x
3
, x
4
, x
5
> 0,
ÒÁ×ÎÏ 130. ôÏÇÄÁ ÎÏ×ÏÅ ÏÐÏÒÎÏÅ ÒÅÛÅÎÉÅ ÂÕÄÅÔ:
x
(2)
1
= 130, x
(2)
2
= 0, x
(2)
3
= 90, x
(2)
4
= 0,
x
(2)
5
= 120, f
(2)
= −1040 < f
(1)
.
ôÅÐÅÒØ Õ ÎÁÓ x
2
É x
4
Ó×ÏÂÏÄÎÙÅ, ÐÒÅÏÂÒÁÚÕÑ (∗) ÐÏ ÆÏÒÍÕÌÁÍ (8), (10),
(11), ÐÏÌÕÞÉÍ
x
1
= 130 −x
2
/2 −x
4
/2,
x
3
= 90 −
3
2
x
2
+
1
2
x
4
,
x
5
= 120 −3x
2
+ 2x
4
,
f = −1040 − 8x
2
+ 4x
4
.
éÚ ÐÏÌÕÞÅÎÎÏÇÏ ×ÙÒÁÖÅÎÉÑ ÄÌÑ f ×ÉÄÎÏ, ÞÔÏ ÔÅÐÅÒØ ÍÏÖÎÏ Õ×ÅÌÉÞÉ×ÁÔØ
x
2
:
x
(3)
1
= 110, x
(3)
2
= 30, x
(3)
4
= x
(3)
5
= 0
84                                                                        3. ëÏÎÔÒÏÌØÎÁÑ ÒÁÂÏÔÁ ½6

  ðÅÒÅÊÄÅÍ Ë ËÁÎÏÎÉÞÅÓËÏÊ ÚÁÄÁÞÅ; ÄÌÑ ÜÔÏÇÏ ××ÅÄÅÍ ÄÏÐÏÌÎÉÔÅÌØÎÙÅ ÐÅ-
ÒÅÍÅÎÎÙÅ x3, x4, x5, ÔÏÇÄÁ:
                         
                           x + 2x2 + x3 = 220,
                          1
                         
                            2x1 + x2 + x4 = 260,
                         
                           4x1 + 5x2 + x5 = 640,
                         
                             x1 > 0, i = 1, . . . , 5,
             f (x1, x2, x3, x4, x5) = −8x1 − 12x2 − 0 · x3 − 0 · x4 − 0 · x5.

       îÁÞÁÌØÎÏÅ ÏÐÏÒÎÏÅ ÒÅÛÅÎÉÅ ×ÏÚØÍÅÍ × ×ÉÄÅ:
                (1)      (1)                (1)               (1)                   (1)
               x1 = x2 = 0,                x3 = 220,        x4 = 260,             x5 = 640,

ÔÏÇÄÁ f = 0.
   ðÅÒÅÍÅÎÎÙÅ x3 , x4, x5 ¡ ÂÁÚÉÓÎÙÅ, Á x1, x2 ¡ Ó×ÏÂÏÄÎÙÅ:

                                          x3 = 220 − x1 − 2x2,
                                          x4 = 260 − 2x1 − x2,                                 (∗)
                                          x5 = 640 − 4x1 − 5x2.

÷ ÆÏÒÍÕ f É x1 É x2 ×ÈÏÄÑÔ Ó ÏÔÒÉÃÁÔÅÌØÎÙÍÉ ËÏÜÆÆÉÃÉÅÎÔÁÍÉ. âÕÄÅÍ
ÉÚÍÅÎÑÔØ x1, ÎÁÉÂÏÌØÛÅÅ ×ÏÚÍÏÖÎÏÅ ÚÎÁÞÅÎÉÅ, ÐÒÉ ËÏÔÏÒÏÍ x3, x4, x5 > 0,
ÒÁ×ÎÏ 130. ôÏÇÄÁ ÎÏ×ÏÅ ÏÐÏÒÎÏÅ ÒÅÛÅÎÉÅ ÂÕÄÅÔ:
                       (2)                   (2)            (2)               (2)
                      x1 = 130,            x2 = 0,         x3 = 90,         x4 = 0,
                                   (2)
                               x5 = 120,             f (2) = −1040 < f (1) .

   ôÅÐÅÒØ Õ ÎÁÓ x2 É x4 Ó×ÏÂÏÄÎÙÅ, ÐÒÅÏÂÒÁÚÕÑ (∗) ÐÏ ÆÏÒÍÕÌÁÍ (8), (10),
(11), ÐÏÌÕÞÉÍ

                                         x1 = 130 − x2/2 − x4/2,
                                                    3     1
                                          x3 = 90 − x2 + x4,
                                                    2     2
                                          x5 = 120 − 3x2 + 2x4,
                                         f = −1040 − 8x2 + 4x4.

       éÚ ÐÏÌÕÞÅÎÎÏÇÏ ×ÙÒÁÖÅÎÉÑ ÄÌÑ f ×ÉÄÎÏ, ÞÔÏ ÔÅÐÅÒØ ÍÏÖÎÏ Õ×ÅÌÉÞÉ×ÁÔØ
x2 :
                             (3)                   (3)              (3)     (3)
                        x1 = 110,             x2 = 30,            x 4 = x5 = 0

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