. , .
, ,
f'x (x 0, y 0) = f'y (x 0, y 0)=0 , . .
z = f (x, y)
. f (x, y) (x 0, y 0), f'x = f'y = 0.
D = f''xx × f''yy (f''xy)2> 0, (x 0, y 0) , , f''xx < 0, , f''xx > 0.
f''xx × f''yy (f''xy)2 < 0, (x 0, y 0) .
, f''xx × f''yy (f''xy)2 = 0 (x 0, y 0), .
. z = xy + +
(x >0, y >0).
◄ , .
= , = x . . x = . y = = y 4 = . y = . 8 y y 4=0, y (8 y 3)=0 y 1=0, y 2=2.
x 1 x 2, y.
x 1 y 1=0 . =2 x = = 5. , M (5; 2).
.
= (y 50 x 2)' x =2×50× x 3 = ;
= (x 20 y 2)' y =2×20× y 3 = ;
= 1. D= × M (5; 2).
D= = = = = 4 1 > 0.
. (5;2) = > 0, . :
z = 5×2 + + =30. ►
D ( ).
. ( ) .
. , .
:
1) D ;
2) D;
3) .
. , , , (. .).
◄ D.
.
,
|
|
, D , . . , , . , , . . , , . . . : .
.
. 0 6, .. . , . .
.
, ,
, , .
.
, .
, . , .
; ; .
: ; . ►
, , u = f (x, y), , .. f (, ) = 0, . , .. . u = f (x, y (x))
. (1)
:
=0.
:
. (2)
(2) l (1):
.
l , :
.
. . .
u = f (x, y) + l f (x, y) .
. f (x, y) = xy, :
2 x + 3 y 5 = 0.
◄ . : :
,
, . ►
.
, , .
III:
11 .
3.11.1 .
:
1. .
2. .
3.