Multiplicadores de Lagrange

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c
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c
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c c
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c / 
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& c



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Ec  c
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c ‫ ݕ‬¹ ͲǤc

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c
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 c
cሺേʹǡͳሻǡ ሺേʹǡ െͳሻǤcc

c  cc  cccc‫ ݕݔ‬ൌ ʹ›‫ ݕݔ‬ൌ െʹc

cc c  (ccc 
cccccc
cc  ccc c
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 c   c
c c c cc
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c
 
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c)3.+c ͳ ‫ ݂׏‬ൌ ࢏ ൅ ʹ࢐ǡ‫ ‰׏‬ൌ ࢏ ൅ ࢐ǡ‫ ݂׏ݕ‬ൌ ʹ‫‰׏‬c ʹ 'c c
c)G3.+c ͳ ‫ ݂׏‬ൌ ࢏ െ ʹ࢐ǡ‫ ‰׏‬ൌ െ ࢏ ൅ ࢐ǡ‫ ݂׏ݕ‬ൌ െʹ‫‰׏‬c ʹ

c / 
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c
 c    c )(+c
c  cc
ccc   "c$ c c   c#c .c)(+cc 3)(+75c 8c .c c 3c c     c c ܸത ‫ͳܩ‬c c    c cܸത ‫ʹܩ‬c c 
 c c  c c  c  c   c c c  c c  
c
   c c  c ߣc c ߤ"c 'c

 cc  (c c
c&)(+ccc (c
c  c  c   c cc 
 c c  cc(cߣccߤc
c
 c c c
 c ܸത ݂ ൌ ߣܸത ‫ ͳܩ‬൅ ߣܸത ‫ʹܩ‬c .c)(+75cc 3)(+75c

c / 
c0"c c c c c c c c c c

´  c ."c  c c.3".5cc.?A0c
c."c'  cc  "c I

 c c  c  ccc)+c7ccc c c c 7c 7

c

c

c $c

c c  c cc
 #c ‫ ݂׏‬ൌ ‫ ݅ݕ‬൅ ‫݆ݔ‬cc ɉ‫ ‰׏‬ൌ ɉ‹ ൅ ɉŒ ൌ ɉሺ‹ ൅ Œሻcc 7ɉc 7ɉcc7c 9  (cccc7c47.5c 37.5c 71c c 
c71c c c 3"c  c c.3".5cc.?A0c
c3"c'  cc  "c I

 c c  c  c 7 7

cc c c c

c

c c

c ‫ ݂׏‬ൌ ʹ‫ ݅ݔ‬൅ ʹ‫݆ݕ‬cc ɉ‫ ‰׏‬ൌ ɉ‹ ൅ ɉŒ ൌ ɉሺ‹ ൅ Œሻcc 37ɉcccc37ɉcc 'c373cc 7c 9  (cc7 $c c47Ac 37Ac cc73cc73c

c

c c ?"c  c c.3".5cc.?A0Jc
c.1"c'  cc  "c I

 c c  c  c 7 7c

c c

cc c c c

c

c ‫ ݂׏‬ൌ ʹ‫ ݅ݔ‬൅ ʹ‫ ݆ݕ‬൅ ʹ‫ݖ‬¥cc ɉ‫ ‰׏‬ൌ ɉ‹ ൅ ɉŒ ൅ ɉœ ൌ ɉሺ‹ ൅ Œ ൅ œሻcc 37ɉ›ʹ› ൌ ɉccccccc3(7ɉc 'c37373(c c
c77(c 9  (cc7 447Fcc ?7Fc 73c 73c (73c

c

c A"c c  
cc  c  c6 G/ cK 3ccC5.c   3."cc c  
c6cc
 
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c)(+c c cc67A55‫ ݖ‬ଶ "c  - c c   
c c c c c cc
  #c‫ ݔ‬ଶ ൅ ‫ ݕ‬ଶ ൅ ‫ ݖ‬ଶ ൌ ͳcc 7‫ ݔ‬ଶ ൅ ‫ ݕ‬ଶ ൅ ‫ ݖ‬ଶ ൌ ͳc

67A55‫ ݖ‬ଶ c c

‫ ܶ׏‬ൌ ͶͲͲ‫ ݖݕ‬ଶ ݅ ൅ ͶͲͲ‫ ݖݔ‬ଶ ݆ ൅ ͶͲͲ‫ݖݕݔ‬¥cc

c

ߣ‫ ݃׏‬ൌ ʹ‫ ݅ߣݔ‬൅ ʹ‫ ݆ߣݕ‬൅ ʹ‫ߣݖ‬¥cc

c

ଶ଴଴


A55‫ ݖ‬ଶ 73ߣc c

ଶ଴଴

A55‫ ݖݔ‬ଶ 7ʹ‫ߣݕ‬c cc










ൌ ߣc

ൌ ߣc

J55(73(ߣcc cA557ߣc ଶ଴଴


ଶ଴଴




c7c







c

‫ ݕ‬ଶ 7‫ ݔ‬ଶ cccc7c

ଶ଴଴

ccccccccccc







ൌ ͶͲͲ‫ݕݔ‬cc

(7ξʹͲͲ7.5ξʹc ccccccccccc‫ ݔ‬ଶ ൅ ‫ ݔ‬ଶ ൅ ʹ‫ ݔ‬ଶ ൌ ͳcc

ccccccccccͶ‫ ݔ‬ଶ ൌ ͳcc ଵ

ଵ

ଵ

cccccccccc‫ ݔ‬ൌ േ cc‫ ݕ‬ൌ േ cc‫ ݖ‬ൌ േ ξʹcc ଶ ଶ ଶ cccccccccc$ cc c  cc c c  ccc

ଵ

ଵ

ଵ

67cA55cLc) ൈ ൈ ሺ ξʹሻሻc ଶ ଶ ଶ cccccccccccc6715ξʹ7050c 1"c  "cI

c c  c  cc)+7cCെ‫ ݔ‬ଶ െ‫ ݕ‬ଶ c c c c4?7.3c 7Cെ‫ ݔ‬ଶ െ‫ ݕ‬ଶ c

7c4?7.3c c

‫ ݂׏‬ൌ െʹ‫ ݅ݔ‬െ ʹ‫݆ݕ‬cc ɉ‫ ‰׏‬ൌ ɉ‹ ൅ ͵ɉŒ ൌcɉሺ‹ ൅ ͵Œሻc G37ɉc cccccccccc

G37?ɉcccccccccccɉ ൌ

ିଶ


& c#cc G37c

ିଶ


ଵ


ଷ

ଷ

c

c

7c cccc7c?c ଷ

9  (cc7c4?7.3c 4?)?+7.3c .57.3c 7."3c 7c?c)."3+c 7?"Fc c)."3?"F+c7cG1"Ac c c !%  )(c  
cc c  c6 cc/ c  
cc c c'  c c c c c c c