Page 167 - DECO403_MATHEMATICS_FOR_ECONOMISTS_PUNJABI

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E L-LOVELY-H math10-1 IInd 21-10-11 IIIrd 24-1-12 IVth 21-4-12 Vth 20-8-12 VIth 10-9-12

fJekJh^10 L T[uso ns/ fBwBso L d' uo ns/ b/Aro/Ai r[De ;fjs gqshpzX T[uso ns/ fBwBso

fJ; soQK, fpzd{ (1, 3) T[Zs/ u fBwBso j't/rk B'N

(1, ^3) d/ ;zdoG ftZu
f xx = 9x = 9 > 0
f yy = 9y = 27 < 0
2
f xx f yy – (f xy ) = 9(–27) – 0 < 0
fJ; soQK fpzd{ (1, ^3) T[Zs/ gbkfJD fpzd{ (Bk tZX s'A tZX bk fBT{Bsw) j't/rk.
(^1, 3) fpzd{ d/ ;zdoG ftZu

f xx = 9x = – 9 < 0
f yy = 9y = 27 > 0
2
f xx f yy – (f xy ) = (–9) (27) – 0 < 0
fpzd{ (^1, 3) T[Zs/ th gbkfJD jZb gqkgs j't/rk.
(^1, ^3) d/ ;zdoG ftZu
f xx = 9x = –9 < 0, f yy = 9y = –27 < 0
2
f xx f yy –(f xy ) = (–9) (–27) – 0 > 0
fpzd{ (^1, ^3) T[Zs/ u dk w[Zb tZX s'A tZX j't/rk
fBopZX d/ Bkb T[uso ns/ fBwBso b/Aro/Ai r[DKe ftXh wzfBnk, T[g:'rsk cbB ns/ nkwdB
fBopZX fBwB fdZsk j?^ u = f (x, y) P x x + P y y = M
fJZE/ u → T[g:'rsk x, y → t;s{nK, M → nkwdB px ns/ py → t;s{nK dh ehws
fJZE/ T[gG'esk nkgDh T[g:'rsk tZX s'A tZX eoBk ukj[zdk j? fdZs/ j'J/ nkwdB fBopZX T[Zs/ sK
b/Aro/Ai r[DKe dk gq:'r eoB s/
v = f (x, y) + λ(M – P x . X – P y . y)

10H2 b/Aro/Ai r[De ;fjs gqshpzX T[uso ns/ fBwBso
(Constrained Maxima and Minima with Langrange’s Multiplier)

b/Aro/Ai ftXh dh ;jkfJsk Bkb th T[jh Bshi/ gqkgs j'Dr/ fijV/ T[go'es, ftXh d[nkok gqkgs j'J/
jB. s[PNheoD cbB ns/ piN o/yk B{z b?D s/
V = f (q 1 q 2 ) + λ(y – p 1 q 1 – p 1 q 2 )
fJZE/ V, λq 1 ns/ q 2 dk cbB j? ns/ λ b/Aro/Ai r[De (Multiplier) j?. fJZE/ ;kvk T[d/P V B{z tZX
s'A tZX eoBk j?. fJ; soQK V B{z q 1 , q 2 ns/ λ ;zdoG ftZu nzfPe fBy/VB (Partial differentiation)
eoe/ Iho' d/ pokpo oZyD s/

...(4)

....(5)

....(6)

;wheoD (4) ns/ (5) B{z b?D s/
f 1 = λp 1 ns/ f = λp 2

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