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f(x, y) = x 2 + y 2
V = {(x, y) ∈ R 2 : x 2 + y 2 + x 2 y 2 = 1}
V g(x, y) = x 2 + y 2 + x 2 y 2 − 1
1 (x, y) ∈ V 0 ≤ x 2 + y 2 =
1 − x 2 y 2 ≤ 1 f
L(x, y, λ) = f(x, y) + λg(x, y) ∇L = 0
⎧
⎪⎨ ∂xf(x, y) + λ∂xg(x, y) = 0,
∂yf(x, y) + λ∂yg(x, y) = 0,
⎪⎩
g(x, y) = 0
⎧
⎪⎨ 2x + λ(2x + 2xy
⎪⎩
2 ) = 0,
2y + λ(2y + 2yx2 ) = 0,
x 2 + y 2 + x 2 y 2 = 1
x = 0 λ = −1/(1 + y 2 ) x = 0 y = ±1
y = 0 λ = −1/(1 + x 2 ) y = 0 x = ±1
x = 0 y = 0 x = ±y 2x 2 + x 4 = 1 x = ± √ 2 − 1
y = ± √ 2 − 1 f(0, ±1) = f(±1, 0) = 1 f(± √ 2 − 1, √ 2 − 1) =
f(± √ 2 − 1, − √ 2 − 1) = 2( √ 2 − 1) 2( √ 2 − 1) < 1 √ 2 − 1 <
1/2 √ 2 < 3/2 2 < 9/4 (0, ±1) (±1, 0)
x 2 =
1 − y2 f , y2 =
1 + y2 1 − y2
1 − y2
,
1 + y2 1 + y 2 + y2 .
0 ≤ t ≤ 1
1 − t
k(t) = + t.
1 + t
k ′ (t) = t2 +2t−1
(t+1) 2 t = √ 2 − 1
t = √ 2 − 1
t = 0 t = 1 t = 0 y = 0
x = ±1 t = 1 y = ±1 x = 0 t = √ 2 − 1 |y| = √ √
2 − 1 |x| = 2 − 1
f(0, ±1) = f(±1, 0) = 1 f(± √ √ √ √ √
2 − 1, 2 − 1) = f(± 2 − 1, − 2 − 1) = 2( 2 − 1)
(0, ±1) (±1, 0)
R 2
V V = {(x, y) ∈ R 2 : x 2 + y 2 + x 2 y 2 = 1}
f(x, y) = x 2 + y 2
f(x, y, z) = x 2 − x + y 2 +
y(z + x − 1) V = {(x, y, z) ∈ R 3 : x 2 + y 2 = 1 ∧ x + y + z = 1}