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γ(t) = (t, 1/t) t → ±∞
L(x, y, λ) = x(x 2 + y 2 ) + λxy ∂xL(x, y, λ) = 3x 2 + y 2 + λy ∂yL(x, y, λ) =
2yx + λx = x(2y + λ) x = 0
y = −λ/2 x = 0 y = −λ/2 λ = 0
x = −2/λ
3 4 λ2 λ2
+ −
λ2 4 2
12 λ2
= 0 ⇐⇒ =
λ2 4 ,
λ = ±2 4√ 3 ±(1/ 4√ 3, 4√ 3)
f(1/ 4√ 3, 4√ 3) = 4/3 3/4 f(−1/ 4√ 3, − 4√ 3) = −4/3 3/4
x = 0
y = 1/x h(x) = f(x, 1/x) = x 3 + 1/x h ′ (x) = 3x 2 − 1/x 2
3x 4 = 1 x = ±1/ 4√ 3 h(1/ 4√ 3) = 4/3 3/4 h(−1/ 4√ 3) = −4/3 3/4
f(x, y) = e x + e y
x + y = 2
g(x, y) = x + y − 2 S = {(x, y) : g(x, y) = 0}
γ(t) = (t, 2 − t)
t → +∞ (xn, y2)
+∞ −∞
fS (x, y) → ∞ (x, y) ∈ S +∞ f L(x, y, λ) =
e x + e y + λ(x + y − 2) ∂xL(x, y, λ) = e x + λ ∂yL(x, y, λ) = e y + λ
e x = e y = −λ
x = y x = y = 1
f(1, 1) = 2e
x = 2 − y
h(y) = f(2 − y, y) = e 2−y + e y .
h ′ (y) = −e 2−y + e y −e 2 + e 2y = 0 y = 1
x = 1 h ′′ (y) = e 2−y + e y h ′′ (1) = 2e > 0 (1, 1)
E a, b, c > 0
g(x, y, z) := x2
− 1 = 0.
a b c2 P [−x, x]×[−y, y]×[−z, z] V (x, y, z) =
8xyz x, y, z ≥ 0 P E (x, y, z) ∈ E
V E ∇g(x, y, z) = (0, 0, 0) x = y = z = 0
(0, 0, 0) /∈ E g(0, 0, 0) = 0
x2 y2 z2
L(x, y, z, λ) = 8xyz + λ + + − 1 .
a2 b2 c2 x, y, z V = 0
xyz > 0
⎧
⎪⎨ ∂xL(x, y, z, λ) = 8yz + 2λx/a
⎪⎩
2
∂yL(x, y, z, λ) = 8xz + 2λy/b2 ∂xL(x, y, z, λ) = 8xy + 2λz/c2 y2 z2
+ + 2 2