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fn g : R 2 → R
g(x, y) = x 2 + 3xy 2 + 2y 4 h(s) = s n h : R → R
f(x, y) = h(g(x, y)).
n n
h n n
f(x1, y1) = h(g(x1, y1)) > h(g(x2, y2)) = f(x2, y2) ⇐⇒ g(x1, y1) > g(x2, y2),
f
g n
n g γ(t) = (0, t)
lim g ◦ γ(t) = +∞ g
t→∞
f
g(x, y) = x + 3
2 y2
2 − 1
4 y4 .
γ(t) = (−3/2t2 , t) lim g◦γ(t) = lim
t→∞
g f
g
∂xg(x, y) = 2x + 3y 2 , ∂yg(x, y) = 6xy + 8y 3 = 2y(3x + 4y 2 )
t→∞ −t 4 /4 = −∞
y y = 0
x x = 0 y = 0 y x =
−4y 2 /3 x −8y 2 /3 + 3y 2 = 0
(−8/3+3)y 2 = 0
g(0, 0) = 0 V γ(t) = (−3/2t 2 , t)
t > 0 γ(t) ∈ V
ε > 0 B((0, 0), ε) ⊆ V
|γ(t)| = 9/4t 4 + t 2 t → 0 δ > 0 |t| < δ
|γ(t)| < ε γ(t) ∈ B((0, 0), ε) ⊆ V
g ◦ γ(t) = −t 4 /4 < 0 = f(0, 0) t ∈]0, δ[ 0
g g(0, 0) γ2(t) = (t, 0)
t γ(t) V
g ◦ γ(t) = t 2 > 0 = g(0, 0) t = 0 (0, 0)
g g(0, 0) g g(0, 0) (0, 0)
g f
(0, 0)
f h 0 (x, y) g(x, y) = 0
f f
g g (0, 0)
n h(s)
s = 0 x 2 + 3xy 2 + y 4 = 0 f
0
h [0, +∞[ ] − ∞, 0]
G + := {(x, y) : x 2 + 3xy 2 + y 4 > 0} g
f g
g(G + ) ⊆]0, +∞[ h g
(0, 0) /∈ G +
g f G − := {(x, y) : x 2 + 3xy 2 + y 4 < 0} f
x 2 + 3xy 2 + y 4 = 0 f 0