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⎧
⎪⎨ 1 + λ(2x − 2) = 0,
1 + 2λy = 0,
⎪⎩
x 2 + y 2 − 2x = 0.
y = 0
λ = −1/(2y) x = 1
λ = −1/(2x − 2) y = x − 1
x 2 +(x−1) 2 −2x = 0 2x 2 −4x+1 = 0 x1 = 1+ √ 2/2
y1 = √ 2/2 x2 = 1 − √ 2/2 y2 = − √ 2/2
f(x1, y1) = √ 2 f(x2, y2) = − √ 2 (x1, y1)
(x2, y2)
V = {(x, y) ∈ R 2 :
x 2 /9 + y 2 = 1}
f f(x, y) := x 2 + y 2 ≥ 0 f
V
3 1
f
F (x, y) = f 2 (x, y) := x 2 + y 2
3 1
ϕ(θ) = (3 cos θ, sin θ) θ ∈
[0, 2π]
F (ϕ(θ)) = 9 cos 2 θ + sin 2 θ = 8 cos 2 θ + 1
d
F (ϕ(θ)) = 16 cos θ sin θ = 8 sin 2θ
dθ
d2 F (ϕ(θ)) = 16 cos 2θ.
dθ2 sin 2θ = 0 θ = 0, π/2, π, 3π/2
θ = 0, π θ = π/2, 3π/2
(±3, 0) 3 (0, ±1)
1
x 2 + y 2 = c 2
x 2 /9 + y 2 = 1
y 2 = c 2 − x 2
x 2 /9 + c 2 − x 2 = 1 8x 2 − 9(c 2 − 1) = 0 c 2 − 1 = 0
x = 0 y = ±1
x 2 = c 2 − y 2 8/9(c 2 − y 2 ) + y 2 = 1 y 2 − (9 − 8c 2 ) = 0
9 − 8c 2 = 0 y = 0 x = ±3
(±3, 0) 3 (0, ±1) 1
(±3, 0) (0, ±1)
g(x, y) = x 2 /9+y 2 −1 ∇g(x, y) = 0 x = y = 0 V = g −1 (0)
(0, 0) /∈ V 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