g(x) = 0.
G ‚" Rn+m
F = (F1, . . . , Fm) : G Rm C1 G
m
S = {(x1, . . . , xn+m) " G : F (x) = 0R }.
x " S
W := det (Fi)x (x ) = 0.

n+j
1d"i,jd"m
U ‚" G x " U V x = (x 1, . . . , x n)
Õ = (Õ1, . . . , Õm) : V Rm C1 (x , Õ(x )) = x
S)"U = {(x, Õ(x)) = (x1, . . . , xn, Õ1(x1, . . . , xn), . . . , Õm(x1, . . . , xn)) : (x1, . . . , xn) " V }.
xn+1 = Õ1(x1, . . . , xn) . . . xn+m = Õm(x1, . . . , xn), (x1, . . . , xn) " V.
F Ck k " N Õ Ck
U ‚" Rn+m f, g1, . . . , gm : U R C1
Å„Å‚
ôÅ‚f(x) = f(x1, . . . , xn, xn+1, . . . , xn+m) min(max)
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚
òÅ‚g1(x) = g1(x1, . . . , xn, xn+1, . . . , xn+m) = 0,
(P )
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚
ôÅ‚g (x) = gm(x1, . . . , xn, xn+1, . . . , xn+m) = 0,
ôÅ‚
ôÅ‚ m
ôÅ‚
ôÅ‚
ółx = (x1, . . . , xn, xn+1, . . . , xn+m) " U.
(P )
 U ‚" Rn+m f, g1, . . . , gm : U R C1
D := {x = (x1, . . . , xn, xn+1, . . . , xn+m) " U : gj(x) = 0, j = 1, . . . , m}.
x " Dd )" D
" r > 0 " x " D )" K(x , r) f(x) e" f(x ),
" r > 0 " x " D )" K(x , r) f(x) d" f(x ),
f x
g1(x) = 0, . . . , gm(x) = 0 f
x g1(x) = 0, . . . , gm(x) = 0
x f
g1(x) = 0, . . . , gm(x) = 0 f
x g1(x) = 0, . . . , gm(x) = 0
x " D
(P ) {"gj(x) : j = 1, . . . , m} m > 1
"g1(x) = 0Rn+1 m = 1

(P )
m
L : Rn+m × Rm ƒ" U × Rm R, L(x, ) = f(x) + jgj(x).
j=1
 (P )
U ‚" Rn+m f, g1, . . . , gm : U R C1
f (P ) x
g1(x) = 0, . . . , gm(x) = 0  " Rm
"L(x ,  ) = 0Rn+2m,
m
n+m
"f(x ) +  j"gj(x ) = 0R , g1(x ) = 0, . . . , gm(x ) = 0.
j=1
(P ) f
Å„Å‚
2
ôÅ‚
òÅ‚x + y2 + z2 ext
(P )1
ôÅ‚
ółx2 + + z2 = 1.
y2
4 9
y2
z2
f(x, y, z) = x2 + y2 + z2 g(x, y, z) = x2 + + - 1 C1
4 9
R3
y2
z2
S = {(x, y, z) " R3 : g(x, y, z) = 0} = {(x, y, z) " R3 : x2 + + = 1} = "

4 9
f
(P )1 (P )1
y
2z
"g(x, y, z) = [2x, , ] = [0, 0, 0] Ô! (x, y, z) = (0, 0, 0) " S.
/
2 9
(P )1
y2
z2
L(x, y, z, ) = f(x, y, z) + g(x, y, z) = x2 + y2 + z2 + (x2 + + - 1).
4 9
(P )1
y2
z2
"L(x, y, z, ) = [0, 0, 0, 0] Ô! [2x + 2x, 2y + 2y , 2z + 2z , x2 + + - 1] = [0, 0, 0, 0] Ô!
4 9 4 9
Å„Å‚
ôÅ‚
ôÅ‚x(1 + ) = 0
ôÅ‚
òÅ‚y(4 + ) = 0
Ô! Ô! (x, y, z, ) " {(Ä…1, 0, 0, -1), (0, Ä…2, 0, -4), (0, 0, Ä…3, -9)}.
2
ôÅ‚z(9 + 2) = 0
ôÅ‚
ôÅ‚
ół
y2
z2
x2 + + - 1 = 0
4 9
f(Ä…1, 0, 0) = 1 f(0, Ä…2, 0) = 4 f(0, 0, Ä…3) = 9 (Ä…1, 0, 0)
f g(x, y, z) = 0
(0, 0, Ä…3) f
g(x, y, z) = 0 (Ä…1, 0, 0)
f g(x, y, z) = 0
(0, 0, Ä…3) f
g(x, y, z) = 0
(0, Ä…2, 0) f
g(x, y, z) = 0 (0, 2, 0) (0, -2, 0)
" µ > 0 " (xµ, yµ, zµ), (x µ, yµ, zµ) " S)"K((0, 2, 0), µ) f(xµ, yµ, zµ) < f(0, 2, 0) < f(x µ, yµ, zµ).
µ " (0, 1) µ " (0, 1)
µ µ2 µ2 3µ
(4, 2 1 - , 0) (0, 2 1 - , ) " S
16 16 4
µ µ2 2 µ2 µ2 µ2
(4, 2 1 - , 0) - (0, 2, 0) = + 4(1 - - 2 1 - + 1) =
16 16 16 16
"
2 2
µ2 µ2 5µ2 µ µ2 5µ
= -3µ + d" -3µ + = Ò! (4, 2 1 - , 0) - (0, 2, 0) d" < µ
16 16 4
µ2 16 2 16
2(1+ 1- )
16
µ2 3µ 2 9µ2 µ2 µ2
(0, 2 1 - , ) - (0, 2, 0) = + 4(1 - - 2 1 - + 1) =
16 4 16 16 16
"
5µ2 µ2 5µ2 µ2 13µ2 µ2 3µ 13µ
= + d" + = Ò! (0, 2 1 - , ) - (0, 2, 0) d" < µ
16 16 4 4
µ2 16 2 16
2(1+ 1- )
16
µ µ2 µ2 µ2 3µ2
f(4, 2 1 - , 0) = + 4(1 - ) = 4 - < 4 = f(0, 2, 0)
16 16 16 16
µ2 3µ 9µ2 µ2 5µ2
f(0, 2 1 - , ) = + 4(1 - ) = 4 + > 4 = f(0, 2, 0).
16 4 16 16 16
(0, Ä…2, 0) f
g(x, y, z) = 0
(P )1 f
g(x, y, z) = 0
R2
U ‚" R2 f, g : U R C2 (x , y ) " U
Å„Å‚
ôÅ‚
òÅ‚f(x, y) ext
(P )2
ôÅ‚
ółg(x, y) = 0.
(x , y ,  ) L(·, ·, ·)
(P )2
Lxx(x , y ,  ) Lxy(x , y ,  ) gx(x , y ,  )
det HL(x , y ,  ) := Lyx(x , y ,  ) Lyy(x , y ,  ) gy(x , y ,  ) = 0,

gx(x , y ,  ) gy(x , y ,  ) 0
f (x , y ) g(x, y) = 0
det HL(x , y ,  ) > 0 f (x , y )
g(x, y) = 0
det HL(x , y ,  ) < 0 f (x , y )
g(x, y) = 0