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∂xz(0, 0) = ∂yz(0, 0) = 0 ∂xf(0, 0, 0) = ∂yf(0, 0, 0) = 0
∂zf(0, 0, 0) = 1
∂ 2 xxz(0, 0) = ∂xf(0, 0, 0) ∂xz(0, 0)∂2 zzf(0, 0, 0) + ∂2 xzf(0, 0, 0)
[∂zf(0, 0, 0)] 2
+
− ∂zf(0, 0, 0) ∂xz(0, 0)∂2 xzf(0, 0, 0) + ∂2 xxf(0, 0, 0)
[∂zf(0, 0, 0)] 2
= −2,
∂ 2
∂yz(0, 0)∂
xyz(0, 0) =
2 zzf(0, 0, 0) + ∂2 yzf(0, 0, 0) ∂xf(0, 0, 0)
[∂zf(0, 0, 0)] 2
+
− ∂zf(0, 0, 0) ∂yz(0, 0)∂2 xzf(0, 0, 0) + ∂2 xyf(0, 0, 0)
[∂zf(0, 0, 0)] 2
= 0,
∂ 2 yyz(0, 0) = ∂yf(0, 0, 0) ∂yz(0, 0)∂2 zzf(0, 0, 0) + ∂2 yzf(0, 0, 0)
[∂zf(0, 0, 0)] 2
+
− ∂zf(0, 0, 0) ∂yz(0, 0)∂2 yzf(0, 0, 0) + ∂2 yyf(0, 0, 0)
[∂zf(0, 0, 0)] 2
= −2.
z (0, 0)
(0, 0) z(x, y)
H(z)(0, 0) =
−2 0
0 −2
,
Γ (x, y) ∈ R 2
y 3 − xy 2 + x 2 y = x − x 3 .
Γ ϕ : R → R
ϕ
ϕ ϕ
x → ±∞
f(x, y) = y 3 − xy 2 + x 2 y − x + x 3 Γ = {(x, y) : f(x, y) = 0}
f(−x, −y) = −f(x, y) Γ
f(0, y) = 0 y = 0 f(x, 0) = 0 −x + x 3 = 0 x ∈ {0, ±1}
f f ∈ C ∞
∂xf(x, y) = −y 2 + 2xy − 1 + 3x 2 = 4x 2 − (y − x) 2 − 1
∂yf(x, y) = 3y 2 − 2xy + x 2 = 2y 2 + (x − y) 2
∂yf(x, y) x = y = 0 x = 0
x γ C 1 x = 0
f(0, y) = 0 y = 0 ϕ(0) = 0
ϕ {(xn, yn)}n∈N Γ xn → 0
0 = lim inf
n→∞ f(xn, yn) = lim inf
n→∞ y3 n,
0 = lim sup
n→∞
f(xn, yn) = lim sup y
n→∞
3 n,
yn → 0 = limn→∞ ϕ(xn) ϕ ϕ(0) = 0 ϕ(±1) = 0
ϕ ∈ C 1 R \ 0 f (0, 0)
df(0, 0) = − dx C ∈ R C + x = 0 γ
(0, 0) x = 0 ϕ 0 x = 0
ϕ ′ (x) = − ∂xf(x, ϕ(x))
∂yf(x, ϕ(x)) = −4x2 − (ϕ(x) − x) 2 − 1
2ϕ2 ,
(x) + (x − ϕ(x)) 2