Monopolio EJERCICIOS RESUELTOS I

 

MIMRKJKJH[ RM[]MAVH[.

7.- ]e chehphajstg khe ueg nuekjûe nuekjûe fm khstms thtgams jjluga luga g K(x) 3 60x gbgstmkm g ue cmrkgfh kuyg nuekjûe fm fmcgefg ms x 3 700 ‛ P/2 G) Hbtme Hbtmelg lg ma mqujajb mqujajbrjh rjh fma chehpha chehphajstg. jstg. B) Mxprmsm Mxprmsm ma JCl me nuekjûe nuekjûe fma prmkjh prmkjh y ag magstjkjfgf magstjkjfgf prmkjh prmkjh fm ag fmcgefg. K) Hbtme Hbtmelg lg ag rmagkjûe qum mxjstm mxjstm me ma mqujajbr mqujajbrjh jh metrm ag magstjkjfg magstjkjfgf f fm ag fmcgefg y ma cgrlme prmkjh-khstm cgrljega. [haukjûe. J. ]e chehp hehpha hajjstg stg qum qum cg cgxj xjcj cjzg zg ma bmem bmemnnjkj kjh h sm me mennrm rmet etgg ga sj sjlu lujm jmet etmm  prhbamcg9 Cgx9  

B(x) 3 J(x) ‛K(x)  phr ah tgeth B(x) 3 p(x)x ‛ K(x)

Ag kh khef efjk jkjû jûe e fm prjc prjcmr mr hrfme hrfme fma prhbam prhbamcg cg fm hptjcj hptjcjzgkj zgkjûe ûe fm kugaqu kugaqujmr jmr mcprms mcprmsgg cgxjcjzgfhrg fm bmemnjkjh jcpajkg9 JCl 3 KCl

ζB/ζx 3 ζJ(x)/ζx ‛ ζK(x)/ζx 3 JCl ‛ KCl 3 0 ↖

JCl 3 KCl

Ma JCl cjfm cjfm kûch sm chfjnjkg ma ejvma fm jelrmshs fm ag mcprmsg kugefh êstg jekrmcmetg ag prhfukkjûe me ueg uejfgf y pumfm mxprmsgrsm khch9

JCl 3 ζJ(x)/ζx [jmefh JV 3 p*x? sm tjmem methekms9 JCl 3 ζp(x)/ ζp(x)/ ζx * x + ζx/ζx * p(x) JCl 3 p(x) + ζp(x)/ ζx * x

Gquã sm hbsmrvg qum ma JCl tjmem fhs khcphemetms9 ma prmkjh pglgfh phr kgfg uejfgf gfjkjhega y ag vgrjgkjûe fma jelrmsh kugefh sm chfjnjkg ag kgetjfgf prhfukjfg. Hbsmrvm qum kugefh ma prmkjh prmkjh eh sm chfjnj chfjnjkg kg getm vgrjgkjh vgrjgkjhems ems fm ag kgetjfgf kgetjfgf hnrmkjfg hnrmkjfg (ζp/ζx (ζp/ζx 3 0), 0), JCl 3 P phr ah qum ag mcprmsg ms prmkjh gkmptgetm.

ma

 

[j ag mcprmsg ms chehpûajkg, ma JCl ms ueg nuekjûe fma ejvma fm prhfukkjûe. Me mstm kgsh ag mcprmsg tjmem ueg kurvg fm cmrkgfh khe pmefjmetm emlgtjvg (ζp(x)/ζx 1 0)? phr tga rgzûe sj gucmetg ag kgetjfgf fmcgefg y ma chehphajstg rmsphefm hnrmkjmefh msg kgetjfgf, dgbrç methekmss ueg rmfukkjûe methekm rmfukkjûe me ma prmkjh. Phr mstg mstg rgzûe ma JCl fma chehphajstg chehphajstg ms sjmcprm sjmcprm jenmrjhr ga prmkjh (JCl 1 Px) y pumfm thcgr vgahrms emlgtjvhs. Ag khefjkjûe fm smluefh hrfme  fm cgxjcjzgkjûe fma bmemnjkjh rmqujmrm9

ζ2B(x)/ ζx2 3 ζ2J(x)/ζx2 ‛ ζ2K(x)/ζx2  ζJCl/ζx ‛ ζKCl/ζx 1 0 ζJCl/ζx 1 ζKCl/ζx   Msth jcpajkg qum ag fjnmrmekjg fm ags pmefjmetms fm ags kurvgs fm JCL y KCl smg emlgtjvg. Khch ma JCl tjmem pmefjmetm emlgtjvg mstg khefjkjûe sm kucpam sjmcprm qum ma KCl smg khestgetm h krmkjmetm. [j mstm ms fmkrmkjmetm, su pmefjmetm fmbm smr cmehr me vgahr gbshauth g ag fma JCl. Ga jluga qum me khcpmtmekjg pmrnmktg me ma chehphajh fmbm kucpajrsm ag khefjkjûe fm vjgbjajfgf mkheûcjkg. Khe ahs fgths fma prhbamcg, yg qum eh mxjstme khsths njihs, ma bmemnjkjh qum hbtmelg ma chehphajstg fmbm smr cgyhr h jluga qum kmrh (B ≭ 0). Pgrg mxprmsgr mxprmsgr ma jelrmsh fma chehphajh chehphajh me nuekjûe nuekjûe fm ag prhfukkjûe prhfukkjûe ms emkmsgrjh emkmsgrjh hbtmemr  me prjcmr aulgr ag nuekjûe jevmrsg fm ag fmcgefg fm cmrkgfh9 Nuekjûe fm fmcgefg fm cmrkgfh9

x 3 700 ‛ P/2

x + P/2 3 700 → P/2 3 700 ‛ x → P 3 200 ‛ 2 x   Nuekjûe jevmrsg fm ag fmcgefg9

P 3 200 ‛ 2 x

Ma jelrmsh fma chehphajstg vmefrãg fgfh phr9

J(x) 3 P(x)*x

J(x) 3 (200 ‛ 2x)x 3 200x ‛ 2x2 Pgrg cgxjcjzgr ma bmemnjkjh sm tjmem methekms9 Cgx9 B(x) 3 200x ‛ 2x2 ‛ 60x  

J(x)

K(x)

 

Ag khefjkjûe fm 7mr hrfme jcpajkg ag jlugafgf fma jelrmsh cgrljega y ma khstm cgrljega (JCl 3 KCl).

JCl 3 ζJ(x)/ζx 3 ζ(200x -2x2)/ζx 3 200 ‛ 6x

JCl 3 KC K Cl

KCl 3 ζK(x)/ζx 3 ζ(60x)/ζx 3 60  

200 ‛ 60x 3 60

↖ x 3 60

Phr tgeth, ag prhfukkjûe fm mqujajbrjh fma chehphajstg ms x 3 60

Htrg cgemrg fm dgkmr ma mimrkjkjh9

Ma vgahr fma mqujajbrjh sm phfãg hbtmemr fmrjvgefh fjrmktgcmetm ag nuekjûe fm bmemnjkjh khe rmspmkth g x9 ζB(x)/ ζx  3 _ζJ(x) ‛ ζK(x)S/ζx 0 3 ζ_(200x ‛ 2x) ‛ 60xS/ζx 3 200 -6x ‛ 60 → x 3 60 [ustjtuymefh me ag nuekjûe jevmrsg fm ag fmcgefg, sm hbtjmem ma prmkjh fm mqujajbrjh. P 3 200 ‛ 2x 3 200 - 2(60) 3 720

Pumsth qum ma KCl ms khestgetm (ζK(x)/ζx 2 3 0), ag Khefjkjûe fm [mluefh Hrfme sm kucpam  pgrg ahs fgths fma prhbamcg9 ζ2B(x)/ ζx2 3 ζ2J(x)/ζx2 ‛ ζ2K(x)/ζx2  3 - 6 1 0 Ma chehphajstg hbtjmem ue bmemnjkjh me mqujajbrjh fm9  

B 3 720(60) ‛ 60(60) 3 ;.200

  

P, JCl, KCl

 

200

 

720

M

KCl 3 60

  60

JJ.

JCl

Ma jelr jelrmsh msh cgrlje cgrljega ga pumf pumfmm mxprm mxprmsgr sgrsm sm me me nuekj nuekjûe ûe fma fma prmkjh prmkjh y fm ag magstjkjfgf prmkjh fm ag fmcgefg. JCl 3 ζJ(x)/ζx JCl 3 p(x) + x ζp(x)/ζx JCl 3 p(x) (7 + x/p(x) ζp(x)/ζx Ma smluefh têrcjeh fma pgrêetmsjs ms ag jevmrsg fm ag magstjkjfgf fm ag fmcgefg, phr ah qum ma JCl pumfm fmnjejrsm khch9 JCl 3 ζJ(x)/ζx JCl 3 p(x)(7+ 7/M p)

ζJ(x) 3 Px ζx + x ζPx 3 Px + x ζPx ζx ζx ζx ζx Khch ah qum qumrmchs ms mxprmsgr ma jelrmsh cgrljega me nuekjûe fma prmkjh y ag magstkjfgf ph phfm fmch chss meh mehek ekms ms vg vgam amre rehs hs fm ue ueg g hpmr hpmrgkj gkjûe ûe cgm cgmcç cçtk tkg g pg pgrg rg ah ahlrg lrgra rah? h? meh mehek ekms ms Cuatpajkgchs Cuatpajkgc hs y fjvjfjchs fjvjfjchs ag mxprmsjûe mxprmsjûe phr Px/Px. Px/Px. Px Px

  P 2x + Px x ζPx Px

Px

ζx

Px + Px x ζPx Px

Px

Px + x ζPx ζx Px 7 + x * ζPx Px ζx

JCl 3 Px (7 + 7/Mx,p)

 

H tgcbjêe? khch dgbjtugacmetm ag magstjkjfgf prmkjh fm ag fmcgefg ms emlgtjvg9

JCl ₁ Px

( 7-

7

)

Mp

Ma JCl smrç cgyhr, jluga h cmehr qum kmrh me nuekjûe fma vgahr fm ag magstjkjfgf fm ag fmcgefg fm cmrkgfh. 50 sj |Mp| 57   JCl

30 sj |Mp| 37   10 sj |Mp| 17

Khch ma chehphajstg cgxjcjzg bmemnjkjhs fhefm ma jelrmsh cgrljega ms jluga ga khstm cgrljega y fgfh qum êstm sjmcprm ms eh emlgtjvh, me ma mqujajbrjh ma jelrmsh cgrljega tgcphkh pumfm smr emlgtjvh. Gfmcçs, sjmcprm qum ma khstm cgrljega eh smg euah, ms fmkjr  smg phsjtjvh, phsjtjvh, ma mqujajbrjh mqujajbrjh fm ag mcprmsg chehphajstg chehphajstg sm prhfukm prhfukm me ma trgch maçstjkh maçstjkh (|Mp| 57) fm ag kurvg fm fmcgefg. Me têrcjehs fm ahs fgths fma prhbamcg, khch ma khstm cgrljega ms phsjtjvh (KCl 3 6), ag magstjkjfgf fm ag fmcgefg me ma mqujajbrjh ms cgyhr qum ueh me vgahr gbshauth9

M p

ζx ₁ Px

₁ -

ζP x

7

72 0 ₁ -

2

60

7, =

→

| Mp|

₁

7, = 5 7

Ma JCl me ma mqujajbrjh ms jluga g 60 JCl

₁ Px ( 7 +

7 ) n Mp

720 (7

+

7

) 3 60

-7,=

JJJ. Me ma mquja mqujajb jbrj rjh h mxjst mxjstm, m, gf gfmc mcçs çs,, ue uegg rm rmag agkj kjûe ûe metrm metrm ag ma mags gstj tjkj kjfg fgf f fm ag fmcgefg fma cmrkgfh y ma cgrlme prmkjh-khstm cgrljega. Ma mqujajbrjh fma chehphajstg pumfm mxprmsgrsm khch9

ǀMpǀ-7)KCl JCl ≧ Px (7 + 7/Mp) ≧ Px (7 ‛ 7/ǀMpǀ) 3 KCl → Px 3 (ǀMpǀ/ ǀMpǀ-7)KCl

 

Rmhrfmegefh têrcjehs Px - KCl 3 Px

7

≧

Mp

7 ǀMpǀ

Ma agfh jzqujmrfh fm ag mxprmsjûe ms ma ãefjkm fm Amremr h ma Lrgfh fm Phfmr fm Cmrkgfh fma Chehphajstg9 mstm rmnarig ma cgrlme prmkjh-khstm cgrljega h ag fjvmrlmekjg metrm ma prmkjh qum mstgbamkm ma chehphajstg y su khstm cgrljega. Ma cgrlme mstg jevmrsgcmetm rmagkjhegfh khe ag magstjkjfgf magstjkjfgf fm ag kurvg fm fmcgefg. Pgrg ueg nuekjûe fm fmcgefg khe magstjkjfgf magstjkjfgf khestgetm, ma cgrlme fma chehphajstg ms khestgetm.

2.- ]e chehphajstg khe ueg nuekjûe fm khstms khstms thtgams jluga g K(x) 3 x 2 gbgst gbgstmkm mkm g ue cmrkgfh cmr kgfh kuyg kuyg nuekjûe nuekjûe jevmrsg jevmrsg fm fmcgefg fmcgefg ms p 3 ;00 ‛ 6x. Kgakua Kgakuamm ag pêrfjfg pêrfjfg fm mnjkjmekjg qum sunrm mstg mkhehcãg rmspmkth g ueg sjtugkjûe fm khcpmtmekjg pmrnmktg. [haukjûe.

Ag cgxjcjzgkjûe fm bmemnjkjh phr pgrtm fm ue chehphajstg jcpajkg JCl 32 KCl Ma jelrmsh fma chehphajstg ms9 J(x) 3 p(x) x 3 (;00 ‛ 6x) x 3 ;00x ‛ 6x  Phr tgeth ma JCl ms9 JCl 3 fJ(x)/fx 3 f(;00x ‛ 6x 2)/fx 3 ;00 - 4x Fm ag nuekjûe fm khsths sm fmrjvge ahs khsths cgrljegams cgrljegams KCl 3 fK(x)/fx 3 2x Ma mqujajbrjh fma chehphajstg vjmem fgfh phr9 p hr9 JCl 3 KCl

→

;00 ‛ 4x 3 26

→

x 3 ;00/70 3 ;0

Gdhrg sm sustjtuym ma vgahr fm x me ag nuekjûe jevmrsg fm ag fmcgefg P3 ;00 ‛ 6x 3 ;00 ‛ (6 * ;0) 3 740 Gdhrg, ma bmemnjkjh fma chehphajstg me mqujajbrjh ms9 B 3 J(x) ‛ K(x) 3 (740*;0) ‛ ;02 3 6.=00 Lrçnjkgcmetm ag  prhfukkjûe fm mqujajbrjh khrrmsphefm ga pueth fm khrtm fm ags kurvgs fm JCl y KCl. Ma prmkjh fm mqujajbrjh vjmem fmtmrcjegfh phr ag kurvg fm fmcgefg pgrg msm ejvma fm prhfukkjûe.

 

;.- ]e chehphajstg phsmm ag sjlujmetm nuekjûe fm fmcgefg9 x 3 =0 ‛ p? ueg nuekjûe fm khsth thtga? KV 3 20+ 6x. Kgakuagr9 prmkjh y kgetjfgf fm mqujajbrjh y mqujajbrjh, bmemnjkjh h pêrfjfg y lrgnjkgr [haukjûe. Ag cgxjcjzgkjûe cgxjcjzgkjûe fm bmemnjkjh vjmem fgfg phr JCl 3 KCl

Ma jelrmsh fma chehphajstg ms9

J(x) 3 p(x) x

Phr tgeth sm fmbm dgaagr ag nuekjûe jevmrsg fm ag fmcgefg? msth pmrcjtjrç mxprmsgr ma jelrmsh fma chehphajstg me nuekjûe fm ag prhfukkjûe.  

Fm ag nuekjûe fm fmcgefg9 x 3 =0 ‛ p fmspmigchs ma prmkjh? phr tgeth  p +x 3 =0

[gbmchs qum9 sjlujmetm cgemrg9

→

B 3 J(x) ‛ K(x) J(x) 3 p * q

 p 3 =0 - x

phr ah tgeth, phfmchs dgaagr ma J(x) fm ag

sustjtuymefh tmemchs9

J(x) 3 (=0 ‛ x)* x? sjmefh x ags 2

kgetjfgfms fma bjme, sm hbtmejmefh khch rmsuatgfh

J 3 =0x ‛ x

Pgrg kgakuagr kgakuagr ma bmemnjkjh h pmrfjfg fmbmchs rmshavmr9

B 3 J(x) ‛ K(x)? B 3 (=0x ‛ x 2) ‛ 

(20 + 6x)

Khch sm dg vjsth, ag khefjkjûe fm 7mr hrfme jcpajkg? JCl 3 fJ(x)/f(x) 3 f(=0x ‛ x 2)fx 3 =0 ‛ 2x KCl 3 fKV(x)/fx 3 f(20+ 6x)fx 3 6

 

JCl 3 KCl =0 ‛ 2x 3 6

→

x 3 2;