z = f (x; y) 0(x 0; y 0).
0(x 0; y 0) z = f (x; y), 0, (x; y) : f (x; y) ≤ f (x 0; y 0).
0(x0; y 0) z = f (x; y), 0, (x; y) : f (x; y) ≥ f (x 0; y 0).
.
.
1 ( ).
z = f (x; y) 0 (x 0; y 0) , 0:
z'x (x 0; y 0) = z'y (x 0; y 0) = 0.
. , .
, 0 , . .
2 ( ). z = f (x; y) (x 0; y 0) :
,
1) ∆ > 0, 0 z = f (x; y) , < 0 , > 0 ;
2) ∆ < 0, 0 ;
3) ∆ = 0, 0 , . .
.
z = x 2 + xy + y 2 2 x 3 y.
z'x = 2 x + y 2,
z'y = x + 2 y 3.
2 x + y 2 = 0
x + 2 y 3 = 0
y = 4/3, x = 1/3.
= 2 = A, = 1 = B, z''yy = 2 = C .
, .
z = f (x; y) g (x; y) = 0. z = f (x; y) ,
g (x; y) = 0 .
.
g (x; y) .
:
1)
g (x; y) = 0 y = φ(x) z = f (x; y) = f (x;φ(x)), .
, .
.
z = x 2 3 xy+ 12 x 6
2 x 3 y = 0.
y = 2 2/3 x, z = x 2 6 x + 2 x 2 + 12 x = 3 x 2 + 6 x.
z' = 6 x + 6 = 0, x = -1, y = 8/3.
, (-1;8/3) .
2)
:
L (x; y; λ) = f (x; y) + λ g(x; y).
|
|
, λ .
z = f (x; y) L (x; y; λ). :
.
z= x 2 3 xy + 12 x 6 2 x 3 y = 0.
:
L (x; y; λ) = x 2 3 xy + 12 x + λ(6 2 x 3 y)
, :
, (-1;8/3) .
, . , , . , , , . , , .
(x 0; y 0) , f (x 0; y 0) . (x; y) (x 0; y 0) .
f (x; y) < f (x 0; y 0), (x 0; y 0) ; f (x; y) > f (x 0; y 0), (x 0; y 0) .
- , , .
1
:
y |
x |
-1 |
-1 |
2
x 2 1,94, y 2 2,06.
: 0,24.
3
: .
4
.
:
: ;
.
5
.
:
1)
.
.
2)
.
3)
.
4)
.
.
: z , .
3.2