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λ, µ ∈ R
f(x, y) = x 3 + xy + λx + µy
P = (1/ √ 3, 0) f(x, y)
λ µ
λ, µ ∈ R C 2
∂xf(x, y) = 3x 2 + y + λ,
∂yf(x, y) = x + µ.
(1/ √ 3, 0) µ = −1/ √ 3 λ = −1
f(x, y) = x 3 + xy 2 − x − 1
√ 3 y,
∂xf(x, y) = 3x 2 + y − 1 ∂yf(x, y) = x − 1/ √ 3
P
∂ 2 xxf(x, y) = 6x, ∂ 2 yyf(x, y) = 0, ∂ 2 xyf(x, y) = 1,
D 2 f(P ) =
2 √ 3 1
1 0
λ 2 − 2 √ 3λ − 1 = 0 √ 3 ± 2
α ∈ R
.
f(x, y, z) = cos 2 x + y 2 − 2y + 1 + αz 2 .
f
C 2 R 3 γ(t) = (0, t, 0)
lim f ◦ γ(t) = +∞
t→+∞
α < 0 γ(t) = (0, 0, t) lim f ◦ γ(t) = −∞ α < 0
t→+∞
α ≥ 0
f(x, y, z) ≥ y 2
π
− 2y + 1 = f + kπ, y, 0 , k ∈ Z,
2
y ↦→ y2 − 2y + 1 y = 1 ( π
2 + kπ, 1, 0) k ∈ Z
f f( π
2 + kπ, 1, 0) = 0 k ∈ Z