Imave periods ekuTvnis evklides amocana

1. mokle reziume

vTqvaT, yoveli , aris uwyvetad warmoebadi funqcia. maSin, Semdeg eqstremalur amocanas

(4.6)

ewodeba gluvi amocana tolobis tipis SezRudevebiT. aq, aris miznis funqcia, xolo

aris dasaSvebi simravle. cxadia, M aris Caketili. ganmartebis Tanaxmad, aris lokaluri minimali (4.6) –Si (e.i. ), Tu romelime ricxvisaTvis

(4.6) amocanis monacemebisagan Sedgenili wrfivi kombinacia atarebs lagranJis funqciis saxels da aRiniSneba ase:

,

sadac , , _ebs ewodebaT lagranJis mamrav-lebi, _s ewodeba lagranJis mTavari mamravli. uSualod ganmartebe-bidan gamomdinareobs, rom (4.6) amocanis dasaSveb simravleze miznis funqcia da lagranJis funqcia erTmaneTis tolia mudmivi Tanamamravlis sizustiT, anu

, roca

am paragrafSi Cven davasabuTebT, rom tolobis tipis SezRudvebiani gluvi amocanis gamokvleva unda moxdes Semdegi TanmimdevrobiT:

1. 
   maTematikuri modelis Sedgena, anu amocanis formulireba (4.6) saxiT;
   
2. 
   lagranJis funqciis Sedgena (4.6) amocanis monacemebis mixedviT: ;
   
3. 
   (4.6) eqstremalur amocanaSi kritikuli wertilebis (anu iseTi wertilebis, romlebic SesaZloa warmoadgendnen (4.6) –is amonaxsns) gansazRvris mizniT gantolebaTa sistemis Sedgena
   

(4.7)

1. 
   (4.6) amocanisaTvis kritikuli wertilebis simravlis ganisazRvra:
   

;

gadamwyveti mniSvneloba aqvs, rom

1. 
   kritikuli wertilebis gaamokvleva maTgan eqstremumis wertilebis amorCevis mizniT.
   

SevniSnoT, rom amocanis gamokvleva moxdeba igive sqemiT, Tu yvela funqcia: miznis funqcionali da SezRudvebi gansazRvrulebi arian raime Ria simravleze.

2. maTematikuri modelis Sedgena

sailustraciod, ganvixiloT piradi moxmarebis neoklasikuri amocana. vTqvaT, kerZo pirma an firmam (zogadad momxmarebeli) piradi moxmarebis mizniT unda SeiZinos n sxvadasxva dasaxelebis produqtis raodenobebi ise, rom sargeblianobis funqciam, romelic aRwers momxmareblis gemovnebas, miiRos maqsimaluri mniSvneloba. simartivisTvis vgulisxmoT, rom TiToeuli saxis produqtis erTeuli aris usasrulod danawilebadi (praqtikulad aseTia 1 kg. Saqari an milioni wyvili fexsacmeli) da rom valis aReba SeuZlebelia. es ori daSveba garantias gvaZlevs, rom yoveli . cnobilia produqtebis fasebi da Tanxa I, romelic gaaCnia momxmarebels. formalurad, moxmarebis neoklasikuri amocana aris amocana utolobis tipis SezRudvebiT:

, ( da yoveli ).

Tu davukvirdebiT amocanis Sinaarss, davrwmundebiT rom igi sinamdvileSi ufro martivia. arsebiTia is garemoeba, rom funqcia , romelic aRwers momxmareblis gemovnebas, ver iqneba nebismieri. arsebobs bunebrivi SezRudvebi, romelsac akmayofilebs nebismieri momxmareblis gemovneba. am SezRudvebs moxmarebis TeoriaSi aqsiomebs uwodeben. aRvweroT ori maTgani.

1. 
   Tu da , maSin anu, Tu erTi produqtis raodenoba imatebs, xolo danarCeni produqtebis raodenoba fiqsirebulia, maSin krebulis sargeblianoba izrdeba. maTematikurad, es niSnavs, rom funqcia zrdadia TiToeuli argumentis mimarT.
   

ekonomikaSi amas uwodeben zRvruli sargeblianobebis dadebiTobis kanons.

1. 
   Tu da , maSin anu, Tu erTi produqtis raodenoba imatebs, xolo danarCeni produqtebis raodenoba fiqsirebulia, maSin im erTi produqtis zRvruli sargeblianoba mcirdeba.
   

ekonomikaSi amas uwodeben kanons zRvruli sargeblianobis klebadobis Sesaxeb.

am Tvisebebidan gamomdinareobs, rom zRvruli sargeblianoba yovelTvis dadebiTia, Tumca klebadia. es niSnavs, rom funqcia zrdadia, magram misi zrdis tempi mcirdeba argumentebis gazrdasTan erTad.

biujetur SezRudvaSi utoloba ar SeiZleba gvqondes, radgan Tu Tanxa darCa, misi daxarjviT SeiZleba raime produqtis mcire dozis SeZena mainc da amgvarad sargeblianobis gazrda. e.i. moxmarebis neoklasikur amocanaSi realurad sruldeba tolobis tipis SezRudva . maSasadame, moxmarebis amocanas aqvs saxe:

, , .

3. kritikuli wertilebis gansazRvra

imisaTvis, rom tolobis tipis SezRudvebian eqstremalur amocanaSi

(4.8)

ganvsazRvroT kritikuli wertilebi, unda CamovayaliboT da davamtkicoT eqstremalurobis pirveli rigis aucilebeli piroba. amisaTvis, winaswar davamtkicoT erTi damxmare faqti, romelsac damoukidebeli mniSvnelobac aqvs.

lema 1. vTqvaT, da funqciebi uwyvetebia da aris lokaluri eqstremumis wertili

(4.9)

amocanaSi. maSin yoveli iseTi , romlisTvisac sruldeba piroba, warmoadgens lokaluri eqstremumis wertils

(4.10)

amocanaSi, sadac .

damtkiceba. vTqvaT, . arsebobs iseTi, rom sruldeba:

(4.11)

radgan g uwyvetia -Si, am -isaTvis arsebobs iseTi, rom



es ki tolobisa da (4.11) –is Tanaxmad niSnavs, rom

,

anu . SemTxveva analogiuria. 

Teorema 1 /eqstremalurobis pirveli rigis aucilebeli piroba/. vTqvaT, aris dasaSvebi wertili (4.8) amocanaSi da yoveli , , uwyvetad warmoebadia -is raRac midamoSi. Tu aris lokaluri eqstremumis wertili (4.8) amocanaSi, maSin arseboben erTdroulad nulis aratoli lagranJis mamravlebi da , iseTebi rom

(4.12)

anu

.

SeniSvna roca , maSin n –ganzomilebiani Teoremis daskvna trivialurad sruldeba. amitom, umetes SemTxvevaSi Tavidanve Tvlian, rom .

damtkiceba. garkveulobisTvis aviRoT . Teorema davamtkicoT induqciiT SezRudvebis raodenobis, anu –is mimarT. rodesac , maSin Toerema samarTliania wina paragrafis Teoremis /eqstremalurobis pirveli rigis aucilebeli pirobis/ ZaliT. axla, dvuSvaT Teoremis samarTlianoba raodenoba SezRudvebisTvis da davamtkicoT –isTvis.

zogadobis SeuzRudavad SegviZlia vigulisxmoT, rom aranulovania (winaaRmdeg SemTxvevaSi aviRebdiT , danarCen mamravlebs ki nulis tols da Teoremis daskvna trivialurad Sesruldeboda) da rom maSin, gantoleba, romelic sruldeba wertilSi mainc, amoixsneba cvladis mimarT. es niSnavs iseTi = funqciis arsebobas, romelic gansazRvrulia wertilis raime midamoSi, amave midamoSi yvelagan sruldeba

(4.13)

da .

Canawerebis simartivis mizniT, gamoviyenoT Semdegi aRniSvnebi:

• 
  _Ti aRvniSnoT kerZo warmoebuli wertilSi, , .
  
• 
  _Ti aRvniSnoT kerZo warmoebuli wertilSi, .
  

maSin, rTuli funqciis gawarmoebis wesis Tanaxmad, (4.13) –is gawarmoeba i–uri cvladis mimarT gvaZlevs:

anu . (4.14)

lema 1 –is ZaliT, (4.8) SegviZlia gadavweroT Semdegi saxiT:

, (4.15)

sadac . (4.15) amocanaSi SezRudvebis raodenoba saSualebas gvaZlevs gamoviyenoT induqciuri daSveba. amitom, arseboben erTdroulad nulis aratoli lagranJis mamravlebi , da, iseTebi rom yoveli -isTvis sruldeba:

. (4.16)

(4.16) igivea, rac:

=

= = 0, (4.17)

sadac

.

cxadia, (4.17) amtkicebs Teoremas. 

radgan

da

amitom Teorema 1, lagranJis funqciis terminebSi, amtkicebs Semdegs: Tu warmoadgens lokaluri eqstremumis wertils (4.8) amocanaSi, maSin arseboben erTdroulad nulis aratoli ricxvebi , iseTebi rom sameuli (sadac ) warmoadgens Semdegi sistemis amonaxsns:

(4.18)

es niSnavs, rom (4.8) sistemis eqstremumis wertilebi unda veZeboT (4.18) –is iseTi amonaxsnebis saSualebiT, romelTa lagranJis mamaravlebi erTdroulad nulis toli ar xdeba. nebismierad aRebuli lagranJis mamravlebisaTvis lagranJis funqcia dasaSveb simravleze (da kerZod eqstremumis wertilebSi) miznis funqciis tolia mudmivi Tanamamravlis sizustiT, xolo Tu lagranJis mamravlebi SerCeulia (4.18) sistemis amonaxsnebis saSualebiT, maSin aseTi lagranJis funqciis kerZo warmoebulebi nulis toli xdeba eqstremumis wertilebSi. (SevniSnoT, rom (4.18) sistemaSi gvaqvs gantoleba da ucnobi) Tu dasaSvebi simravle Ria araa, TviTon miznis funqciis warmoebuli SesaZloa ar gaxdes nulis toli eqstremumis wertilebSi. rogorc vxedavT, lagranJis meTodi (eqstremumis wertilebis gansazRvra (4.18) –is amonaxsnebis saSualebiT) warmoadgens SezRudvaTa moxsnis saSualebas eqstremalur amocanaSi tolobis tipis SezRudvebiT. amitom, Teorema 1 –is pirobebSi, (4.8) amocanisaTvis kritikuli wertilebis simravle ganisazRvreba Semdegnairad:

. 

agebis Tanaxmad,

ganmarteba -s ewodeba regularuli wertili, Tu , , uwyvetad warmoebadi funqciebia -is midamoSi da wrfivad damoukidebeli veqtorebia.

Tu regularuli wertilia, eqstremalurobis aucilebeli piroba SeiZleba ase CamovayaliboT:

Teorema 3 Tu regularuli wertilia, maSin moiZebneba lagranJis mamravlebi, iseTebi, rom .
SeniSvna SevniSnoT, rom (4.12) niSnavs, rom veqtorebi wrfivad damokidebulia, kerZod, Tu , maSin da veqtorebi arian kolinearulebi. Tu regularuli wertilia, maSin wrfivad damoukidebeli veqtorebia da am SemTxvevaSi, (winaaRmdeg SemTxvevaSi veqtorebi wrfivad damoukidebuli gaxdebodnen), amitom Tu regularuli wertilia, SegviZlia CavTvaloT, rom da (4.18) gantolebaTa sistemaSi gantolebebis da saZiebeli cvladebis raodenoba gaxdeba erTmaneTis toli.

Tu ar aris regularuli wertili, e. i. wrfivad damokidebuli veqtorebia, maSin (4.12) gantolebaTa sistemas gaaCnia amonaxsni roca an , orivejer saqme gvaqvs “gadagvarebul” SemTxvevasTan.

ganvixiloT, . am amocanaSi aris erTaderTi dasaSvebi wertili (M gadagvarda wertilSi) da, maSasadame, igi aris amonaxsnic: . igi ar aris regularuli wertili, radgan veqtorebi da am wertilSi wrfivad damokidebulebi arian ((-2,0) da (2,0)). Tu SevadgenT lagranJis funqcias da

axla, ganvixiloT amocana igive SezRudvebiT da sxva miznis funqciiT:

aqac, erTaderTi dasaSvebi wertili ar aris regularuli.

Tu SevadgenT lagranJis funqcias da amovwerT (4.18) sistemas kritikuli wertilis gansazRvrisaTvis, vnaxavT rom mas aqvs amonaxsni -Tvisac, radgan .

magaliTi. amovxsnaT amocana:

amocanas mivceT standartuli saxe: , da . kritikuli wertilebis gansazRvris mizniT SevadginoT sistema:

(4.19)

aq sami gantolebaa oTxi ucnobiT. radgan gantolebebi cvladebis mimarT erTgvarovnebia, amitom cvladebis ricxvis Semcireba SegviZlia ori SesaZlo SemTxvevis cal – calke ganxilvis xarjze.

a). davuSvaT . maSin rac ewinaaRmdegeba mesame gantolebas. e.i. es SemTxveva kritikul wertils ar iZleva.

b). ganvixiloT SemTxveva. SegviZlia -ze gavyoT (4.19)–is pirveli ori gantoleba, ris Semdegac / , SegviZlia CavTvaloT erT cvladad da is aRvniSnoT -iT. (4.19) sistema miiRebs saxes:

(4.20)

Tu SevkrebT I da II gantolebebs, miviRebT , saidanac , an .

jer ganvixiloT . maSin (4.20)-is I da II gantoleba gvaZlevs , xolo III gantolebidan miviRebT: e.i. (4.19) –is amonaxsnebia da radgan , amitom . Tu , wina SemTxvevis msgavsad miviRebT , radgan aris (4.19) –is amonaxsnebi.

axla, ganvixiloT am amocanis geometriuli interpretacia. -s grafiki warmoadgens zedapirs samganzomilebian sivrceSi, xolo SezRudva wrewirs amave sivrceSi (ix. nax.4.1) Ошибка! Ошибка связи.

nax 4.1
gavixsenoT, rom funqciis donis wirebi ewodeba Semdegi saxis simravleebs: . sibrtyeze gamovsaxoT dasaSvebi simravle , romelic warmoadgens erTeulradiusian wrewirs centriT koordinatTa saTaveSi da miznis funqciis donis wirebi, romlebic warmoadgenen hiperbolebs.

Ошибка! Ошибка связи. nax 4.2

Tu warmoebadi funqciaa, maSin gradients aqvs funqciis uswrafesi zrdis mimarTuleba, xolo antigradients uswrafesi klebis mimarTuleba. (ix.8,gv.70). Cvens amocanaSi (ix.nax.4.2) geometriu-lad, am amocanis amoxsna niSnavs dasaSveb simravleze ( wrewirze) vipovoT is wertilebi, romlebic Seesabamebian minimaluri (maqsimaluri) mniSvnelobis mqone donis wirebs. (donis wiris yovel wertilSi miznis funqciis mniSvneloba erTi da igivea)

naxazidan Cans, rom magaliTad, pirvel meoTxedSi, miT metia miznis funqciis mniSvneloba, rac ufro Sors aris donis wiri koordinatTa saTavidan da maT Soris yvelaze Sors myofi da Tan iseTi, romelsac saerTo wertili aqvs dasaSveb simravlesTan, aris is wiri, romelic exeba wrewirs (ix. nax. 4.2). Sexebis wertilis koordinatebi SeiZleba vipovoT Semdegnairad: Sexebis wertilSi hiperbolis da wrewiris mxebebis sakuTxo koeficientebi emTxvevian erTmaneTs. Tu ganvixilavT -s rogorc -s aracxad funqcias da gavawarmoebT da gantolebebs miviRebT

,

Tu maT davumatebT dasaSvebi simravlis ganmsazRvrav gantolebas, saZiebeli wertilisTvis miviRebT Semdeg sistemas.

miviReT igive kritikuli wertilebi
4. kritikuli wertilebis gamokvleva vaierStrasis

Teoremisa da misi Sedegebis gamoyenebiT
es midgoma Zalze zogadia, da amitom ganvixiloT konkretul magaliTze.

magaliTi. /amoxsnis gagrZeleba/. rogorc zemoT miviReT, , anu gvaqvs oTxi kritikuli wertili. simravle gansazRvravs wrewirs, rac kompaqturi simravlea, xolo uwyveti funqciaa. maSasadame, vaierStrasis Teoremis ZaliT, Cvens amocanaSi arsebobs globaluri maqsimalebi da minimalebi.

radgan , amitom K –Si aris rogorc globaluri maqsimali, aseve globaluri minimali. Semdegi Casmebi:

gviCveneben, rom miznis funqcia dasaSveb simravleze Rebulobs mxolod or gansxvavebul mniSvnelobas. amitom, .

5. kritikuli wertilebis gamokvleva uSualod

ganmartebis safuZvelze

gamovikvlioT, magaliTad, . warmovadginoT -is nebismieri wertili saxiT:

anu .

Tu wertili akmayofilebs SezRudvas, maSin agreTve izRudebian, rasac male vnaxavT. jer SevafasoT:

= . (4.21)

(4.21) mxolod im SemTxvevaSi mogvcems saWiro informacias, Tu gaviTvaliswinebT SezRudvas:



,

ris safuZvelzec (4.21) gvaZlevs:

,

e.i.

6. kritikuli wertilebis gamokvleva eqstremalurobis

meore rigis sakmarisi pirobis safuZvelze

kvlav ganvixiloT amocana tolobis tipis SezRudvebiT

sadac funqciebi orjer uwyvetad warmoebadia. aRvniSnoT M –iT dasaSvebi simravle

vidre Teoremas (eqstremalurobis meore rigis sakmaris pirobas) CamovayalibebT da davamtkicebT, moviyvanoT ramdenime martivi faqti.

yoveli sasrulganzomilebiani veqtorisaTvis ganisaz-Rvreba funqcia

romelsac, M simravlis gansazRvris Tanaxmad, aqvs Semdegi mniSvnelovani Tviseba

(4.22)

Tu aris eqstremumis wertili, an Tundac kritikuli wertili, xolo -cali lagranJis mamravli isea SerCeuli, rom veqtori akmayofilebs kritikuli wertilis (4.18) pirobas, maSin

funqcia, garda (4.22) –isa, akmayofilebs agreTve

tolobas.

Teorema 3 /eqstremalurobis meore rigis sakmarisi piroba/. vTqvaT

funqciebs, aqvT uwyveti meore rigis kerZo warmoebulebi, da sruldeba Semdegi ori piroba:

1. 
   , iseTi, rom , sadac
   

1. 
   sruldeba yoveli iseTi aranulovani veqtorisaTvis, romelic akmayofilebs SezRudvebs:
   

maSin, aris mkacri lokaluri maqsimali (minimali) Semdeg eqstremalur amocanaSi: .

damtkiceba. davamtkicoT mkacri lokaluri maqsimalis arseboba. lokaluri minimalis arseboba analogiurad mtkicdeba.

davuSvaT sawinaaRmdego. vTqvaT Teoremis pirobebi sruldeba, magram araa mkacri lokaluri maqsimali. maSin, arsebobs veqtorTa mimdevroba , iseTi rom

(4.23)

da . SemoviRoT aRniSvnebi:

rogorc vxedavT, cxadia, yoveli veqtori Zevs erTeulovan sferoze, romelic kompaqturi simravlea da amitom arsebobs mimdevrobis qvemimdevroba, krebadi romeliRac -isaken. radgan -is norma uwyvetadaa damokidebuli veqtorebze, . zogadobis SeuzRudavad SegviZlia vigulisxmoT, rom TviTon aris krebadi -isaken.

teiloris formulis Tanaxmad, ricxvebisTvis gvaqvs:

. (4.24)

pirobis Tanaxmad, amitom (4.24) igivea, rac

. (4.25)

Tu (4.25) –Si gadavalT zRvarze roca , miviRebT:

anu am -isaTvis SegviZlia gamoviyenoT Teoremis b) piroba, romlis ZaliTac . aRvniSnoT da . radgan , teiloris formula gvaZlevs:

.

radgan , amitom

. (4.26)

radgan , amitom arsebobs nomeri , iseTi rom Cven SegviZlia aviRoT imdenad didi, rom roca , Sesruldes

(radgan ).

sabolood, (4.26) –idan vRebulobT, rom yoveli -isaTvis

rac ewinaaRmdegeba (4.23)–s. 

magaliTi. davasruloT amocanis ganxilva, romelSic . wertilisaTvis gvqonda anu , xolo _araa gansazRvruli arc dadebiTad, arc uaryofiTad; , . ganvsazRvroT veqtorebi, romlebic akmayofileben pirobas.

anu dasasrul, radgan

amitom .

kritikuli wertilebis meore wyvilisaTvis: , gvqonda amitom , da verafers vambobT gansazRvrulobaze. , . Semdeg,

anu CvenTvis saintereso veqtorebs aqvT saxe dasasrul,

amitom . rogorc vxedavT, II rigis pirobebis gamoyeneba mxolod lokalur informacias gvaZlevs kritikuli wertilebis Sesaxeb. miuxedavad amisa, es yvelaze mZlavri meTodia da xSirad gamoiyeneba.

7. lagranJis mamravlebis maTematikuri da ekonomikuri interpretaciebi

ganvixiloT minimizaciis amocana

anu standartuli saxiT:

. (4.28)

cxadia, (4.28) –is yoveli amonaxsni damokidebulia parametr – veqtoris mniSvnelobaze. magaliTad, Tu , maSin amocanas aqvs trivialuri amonaxsni

davuSvaT, , funqciebi aris imdenad kargi, rom b veqtoris yoveli SesaZlo mniSvnelobisaTvis , (4.28) amocanas aqvs erTaderTi amonaxsni , da rom calsaxad ganisazRvrebian lagranJis mamravlebi , romelTaTvisac Teorema 1 –is ZaliT

. (4.29)

radgan TviTon lagranJis funqcia am mamravlebiT aris

,

amitom yoveli x–isaTvis, romelic akmayofilebs (4.28) –is SezRudvebs

, (4.30)

sruldeba

.

kerZod, roca x= , gvaqvs

. (4.31)

gavawarmooT (4.31) –is orive mxare -Ti:

(marjvena mxareSi meore wevri nulis tolia (4.30) –is gamo)

= ((4.29) –is ZaliT) .

amgvarad,

(4.32)

rogorc vxedavT, lagranJis optimaluri mamravlebi gviCveneben, Tu ramdenadaa mgrZnobiare miznis funqciis mniSvneloba SezRudvis parametrebis cvlilebis mimarT. magaliTad, Tu romelime lagranJis mamravli nulis toli aRmoCnda, Sesabamisi parametris mcire cvlileba gavlenas ar iqoniebs miznis funqciis eqstremalur mniSvnelobaze.

lagranJis mamravlebis interpretacia gansakuTrebiT Sinaarsiania ekonomikur amocanebSi. maragis ganawilebis ekonomikur amocanebSi miznis funqcias aqvs Rirebulebis ganzomileba (Rirebuleba aris fasis namravli produqciis moculobaze), xolo SezRudvebis saSualebiT dgindeba produqciis gansazRvruli moculoba. (4.32) –is mixedviT, lagranJis mamravls aqvs fasis ganzomileba. am mizezis gamo, lagranJis mamravlebs uwodeben Crdilovan fass mocemul produqciaze.

8. funqciis uswrafesi cvlilebis mimarTulebis gansazRvra

vTqvaT, da igi Ria simravlea; da aris warmoebadi funqcia -Si. Cveni amocanaa wertilSi funqciis uswrafesi cvlilebis mimarTulebebis gansazRvra. mimarTulebas, Cveulebriv, vuwodebT -is erTeulovani sferos elementebs. mag. Tu aris erT-erTi mimarTuleba (e.i hh^T=1), maSin sakmaod mcire λ>0 ricxvebisaTvis da h mimarTulebiT _is cvlilebis siCqare -Si aris:

radgan f warmoebadia -Si, amitom es sidide aris . amgvarad, Cven gvainteresebs is mimarTulebebi, romlisTvisac iRebs eqstremalur mniSvnelobas.

ganmarteba: wertilSi f-is uswrafesi cvlilebis mimarTulebebi, ganmartebis mixedviT, warmoadgenen Semdegi eqstremaluri amocanis

, (4.33)

eqstremumis wertilebs.

amocana rom aratrivialuri iyos, unda vigulisxmoT ≠0 (aranulovani veqtoria). axla amovxsnaT (4.33).

SevadginoT lagranJis funqcia:

amovweroT sistema kritikuli wertilebis gansazRvrisaTvis:

anu veqtorulad

(4.34)

SevamciroT cvladebis ricxvi SemTxvevaTa ricxvis zrdis xarjze.

1. vTqvaT maSin rac araa Tavsebadi hh^T=1-Tan.

2. ganvixiloT SemTxveva. SegviZlia -ze gavyoT (4.34) –is pirveli gantoleba, ris Semdegac / , SegviZlia CavTvaloT erT cvladad da is aRvniSnoT -iT. (4.34) sistema iRebs saxes (λ=0 gamoiwvevda -s):

,

saidanac,

anu ,

e.i (4.34)-is amonaxsnebia , romlebic gvaZleven or kritikul wertils: K= .

gamovikvlioT es wertilebi. vaierStrasis Teoremis gamoyeneba SegviZlia, radgan erTeulovani sfero kompaqturia, xolo miznis funqcia _ uwyveti. amitom kritikul wertilebs Soris erTi aris globaluri minimali, meore globaluri maqsimali. miznis funqciaSi Casma gvaZlevs, rom gradientis mimarTuleba, anu aris globaluri maqsimali (4.33)-Si, anu f-is uswrafesi zrdis mimarTuleba, xolo antigradientis mimarTuleba aris f-is uswrafesi klebis mimarTuleba Si