Page 165 - DECO403_MATHEMATICS_FOR_ECONOMISTS_PUNJABI

VED1
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

10H1 T[uso ns/ fBwBso gsk eoB d/ gqshpzX B'N
(Conditions to Finding Maxima and Minima)

(A) io{oh gqshpzX (Necessary Condition)L i/eo u = f (x, y)
sK ∂u/∂x = ∂u/∂y = 0
(B) T[fus gqshpzX (Sufficient Condition)L

T[uso d/ bJh, i/eo u = f (x, y) ns/ io{oh gqshpzX fx = 0, and fy = 0 sK
2
2
2
2
∂ u/∂x < 0 and ∂ u/∂y > 0
∂ 2 u ∂ 2 u ∂ 2  u  2
f
2
∴ . >   or AB > C or xx f ⋅ yy > f
xy
xy 
∂ x 2 ∂ y 2  ∂∂
↓ () A ()  () 
C
B

fBwBso d/ bJh
sK

∴

or
2
2
3
T[dkjo 1H u = x + x – xy + y + 4 T[uso ns/ fBwBso w[Zb eZY'.
u ∂
2
jZb L y = 0 ....(1)

= 3x + 2x –
x ∂
u ∂
= –x + 2y = 0 ....(2)
x ∂
(1) s/ (2) B{z jZb eoB s/
2
3x + 2x – y = 0
–x + 2y = 0
i/eo x = 2y sK gfjb/ ;wheoD d/ oZyD s/
2
3(2y) + 2 (2y) – y = 0
y(12y + 3) = 0
1
id'A y = 0 or y = –
4
id'A y = 0
x = 2y = 0
1
id'A y = –
4
1
sK x = 2y = –
2
fJ; soQK, io{oh gqshpzX B{z gsk eoB d/ bJh ;kv/ e'b d' fpzd{ (0, 0) ns/ (^1$2, ^1$4) j?. j[D
n;hA fJj gsk eoKr/ fe T[go'es wkB, T[uso ns/ fBwBso dhnK PowK B{z ;zs[PN eod/ jB iK
BjhA.

LOVELY PROFESSIONAL UNIVERSITY 159