1 Definicja f (x0, y0)
O(x0, y0) (x, y) " O(x0, y0)
f(x0, y0) d" f(x, y).
f (x0, y0)
S(x0, y0) (x, y) " S(x0, y0)
f(x0, y0) < f(x, y).
f (x0, y0)
O(x0, y0) (x, y) " O(x0, y0)
f(x0, y0) e" f(x, y).
f (x0, y0)
S(x0, y0) (x, y) " S(x0, y0)
f(x0, y0) > f(x, y).
2 TWIERDZENIE
f (x0, y0)

2 2
fx(x0, y0) = fy(x0, y0) = 0.


3 Definicja

4 TWIERDZENIE
f (x0, y0)
(x0, y0)

2 2 2 2

fxx(x0, y0) fxy(x0, y0)

W (x0, y0) =
2 2 2 2

fyx(x0, y0) fyy(x0, y0)
2 2 2 2 2 2 2 2
= fxx(x0, y0) · fyy(x0, y0) - fxy(x0, y0) · fyx(x0, y0) > 0,
2 2
f (x0, y0) fxx(x0, y0) > 0
2 2
fxx(x0, y0) < 0

W (x0, y0) < 0 f (x0, y0)

2
2 2 2 2 2 2
W (x0, y0) = fxx(x0, y0) · fyy(x0, y0) - fxy(x0, y0)
W (x0, y0) = 0 f
(x0, y0)
5 Przykład
f(x, y) = x3 + 3xy2 - 15x - 12y,
2 2
fx(x, y) = 3x2 + 3y2 - 15, fy(x, y) = 6xy - 12,
2 2 2 2 2 2 2 2
fxx(x, y) = 6x, fxy(x, y) = 6y, fyx(x, y) = 6y, fxx(x, y) = 6x.

Å„Å‚
2
ôÅ‚
ôÅ‚
x =
òÅ‚
2
fx(x, y) = 0 3x2 + 3y2 - 15 = 0 y =0
y
Ð!Ò! Ð!Ò!
2
22
fy(x, y) = 0 6xy - 12 = 0
ôÅ‚
ôÅ‚
ół + y2 - 5 = 0
y2
4
+ y2 - 5 = 0 Ð!Ò! y4 - 5y2 + 4 = 0 Ð!Ò! (y2 - 1)(y2 - 4) = 0
y2
Ð!Ò! (y - 1)(y + 1)(y - 2)(y + 2) = 0 Ð!Ò! y = 1 (" y = -1 (" y = 2 (" y = -2

2
fx(x, y) = 0 x = 2 x = -2 x = 1 x = -1
Ð!Ò! (" (" ("
2
fy(x, y) = 0 y = 1 y = -1 y = 2 y = -2
(1, 2) (-1, -2) (2, 1) (-2, -1)
W (x0, y0)

2 2 2 2

fxx(x, y) fxy(x, y)
6x 6y

W (x, y) = = = 36(x2 - y2),
2 2 2 2

fyx(x, y) fyy(x, y) 6y 6x

W (1, 2) < 0, W (-1, -2) < 0, W (2, 1) > 0, W (-2, -1) > 0.
f (2, 1) (-2, -1)
2 2 2 2
fxx(2, 1) = 12 > 0, fxx(-2, -1) = -12 < 0,
f (2, 1) f (2, 1) = 8 + 6 - 30 - 12 = -28
(-2, -1) f (-2, -1) = -8 - 6 + 30 + 12 = 28

6 Definicja
F (x, y) = 0

y = y(x)
F (x, y(x)) = 0
x I
7 Przykład
x2 + y2 - 1 = 0,
F (x, y) = x2 + y2 - 1
(x0, y0) = (0, 1) x "
[-1, 1] y = y(x) F (x, y(x)) = 0 y(x0) = y0
x2 + (y(x))2 - 1 = 0 y(0) = 1?


y1(x) = 1 - x2, x " [-1, 1],



1
- x2, x " [-1, 1] )" Q
y2(x) =
- 1 - x2, x " [-1, 1] )" (R \ Q).
y1
(x0, y0) = (1, 0) y = y(x)
x = x(y)
8 TWIERDZENIE (x0, y0) " R2 F
O(x0, y0)
"F "F
O(x0, y0)
"x "y
"F
F (x0, y0) = 0 = 0

"y
O(x0) y = y(x)

F (x, y(x)) = 0 x " O(x0)
y(x0) = y0
"F
(x, y(x))
"x
y2 (x) = - x " O(x0)
"F
(x, y(x))
"y

9 Przykład F (x, y) = x2 + y2 - 1
"F "F
(x, y) = 2x (x, y) = 2y
"x "y
"F
R2 y = 0 (x, y) = 0

"y
(x0, y0) F (x0, y0) = 0 y0 = 0 y

y = y(x)

(x0, y0) = (0, 1) y(x) = 1 - x2
"F
1 -x -x 2x (x, y(x))
"x
y2 (x) = " · (-2x) = " = = - = - .
"F
2 1 - x2 1 - x2 y(x) 2y(x) (x, y(x))
"y
(x0, y0) = (1, 0) x y