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⎧
⎪⎨ 2x + λ2x/9 = 0,
2y + 2λy = 0,
⎪⎩
x 2 /9 + y 2 = 1
y = 0 λ = −1 y = 0 x = ±3
λ = 1 x = 0 y = ±1
(±3, 0) 3 (0, ±1) 1
(±3, 0) (0, ±1)
f(x, y) = (x−1) 2 −y 2
V = {(x, y) ∈ R 2 : x 2 + y 2 = 1}
1 f
V
ϕ(θ) = (cos θ, sin θ) θ ∈ [0, 2π]
f ◦ ϕ(θ) = (cos θ − 1) 2 − sin 2 θ
d
f ◦ ϕ(θ) = 2(cos θ − 1) sin θ − 2 sin θ cos θ = 2 sin θ(1 − 2 cos θ)
dθ
d2 dθ2 f ◦ ϕ(θ) = 4 sin2 θ + 2(1 − 2 cos θ) cos θ.
θ = 0, π, π/3, 5/3π ϕ(0) = (1, 0) f(1, 0) = 0
d2 dθ2 f ◦ ϕ(0) = −2 ϕ(π) = (−1, 0)
f(−1, 0) = 4 d2
dθ2 f ◦ ϕ(π) = −6
ϕ(π/3) = (1/2, √ 3/2) f(1/2, √ 3/2) = −1/2 d2
dθ2 f ◦ ϕ(π/3) = 3
ϕ(5π/3) = (1/2, − √ 3/2) f(1/2, − √ 3/2) = −1/2
d2 dθ2 f ◦ ϕ(5π/3) = 3 (−1, 0)
(1/2, ± √ 3/2)
(x − 1) 2 − y2 = c
x2 + y2 = 1 (x0, y0)
(x0 − 1)(x − x0) − y0(y − y0) = 0
(x0, y0) x0(x − x0) + y0(y − y0) = 0
y0(x0 − 1) + x0y0 = 0 y0 = 0 x0 = ±1
x0 = 1/2 y0 = ± √ 3/2 f(1, 0) = 0 f(−1, 0) = 4
f(1/2, ± √ 3/2) = −1/2 (−1, 0) (1/2, ± √ 3/2)
g(x, y) = x2 + y2 − 1 V = g−1 (0) 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.
⎧
⎪⎨ 2(x − 1) + λ2x = 0,
−2y + 2λy = 0,
⎪⎩
x 2 + y 2 = 1
y = 0 λ = 1 y = 0 x = ±1 f(1, 0) = 0
f(−1, 0) = 4 λ = 1 x = 1/2 y = ± √ 3/2
f(1/2, ± √ 3/2) = −1/2 (−1, 0) (1/2, ± √ 3/2)