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f(x, y, z) = ez+x2 f
+ αx + y 2 − z 2 − 1 f(0, 0, 0) = 0 α
∂xf(x, y, z) = 2xe z+x2
+ α
∂yf(x, y, z) = 2y
∂zf(x, y, z) = e z+x2
− 2z.
∂zf(0, 0, 0) = 1 = 0 f = 0
z = z(x, y) (0, 0) z(0, 0) = 0 f ∈ C 1
C 1
∂xz(x, y) = − ∂xf(x, y, z(x, y)) 2xez(x,y)+x2 + α
= −
∂zf(x, y, z(x, y)) ez(x,y)+x2 − 2z(x, y) ,
∂yz(x, y) = − ∂yf(x, y, z(x, y))
2y
= −
∂zf(x, y, z(x, y)) ez(x,y)+x2 − 2z(x, y) .
∂xz(0, 0) = α ∂yz(0, 0) = 0 (0, 0)
z ∈ C 1 α = 0
α z(x, y) (0, 0)
∂ 2 xxz(x, y) = ∂xf(x, y, z(x, y)) ∂xz(x, y)∂2 zzf(x, y, z(x, y)) + ∂2 xzf(x, y, z(x, y))
[∂zf(x, y, z(x, y))] 2
+
− ∂zf(x, y, z(x, y)) ∂xz(x, y)∂2 xzf(x, y, z(x, y)) + ∂2 xxf(x, y, z(x, y))
[∂zf(x, y, z(x, y))] 2
∂ 2
∂yz(x, y)∂
xyz(x, y) =
2 zzf(x, y, z(x, y)) + ∂2 yzf(x, y, z(x, y)) ∂xf(x, y, z(x, y))
[∂zf(x, y, z(x, y))] 2
+
− ∂zf(x, y, z(x, y)) ∂yz(x, y)∂2 xzf(x, y, z(x, y)) + ∂2 xyf(x, y, z(x, y))
[∂zf(x, y, z(x, y))] 2
∂ 2 yyz(x, y) = ∂yf(x, y, z(x, y)) ∂yz(x, y)∂2 zzf(x, y, z(x, y)) + ∂2 yzf(x, y, z(x, y))
[∂zf(x, y, z(x, y))] 2
+
− ∂zf(x, y, z(x, y)) ∂yz(x, y)∂2 yzf(x, y, z(x, y)) + ∂2 yyf(x, y, z(x, y))
[∂zf(x, y, z(x, y))] 2
f
∂xxf(x, y, z) = 2e z+x2
∂yyf(x, y, z) = 2, ∂yyf(0, 0, 0) = 2
+ 4x 2 e z+x2
, ∂xxf(0, 0, 0) = 2
∂zzf(x, y, z) = e z+x2
− 2, ∂zzf(0, 0, 0) = −1
∂xyf(x, y, z) = ∂zyf(x, y, z) = 0, ∂xyf(0, 0, 0) = ∂zyf(0, 0, 0) = 0,
∂xzf(x, y, z) = 2xe z+x2
, ∂xzf(0, 0, 0) = 0.