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Zhk h, k ∈ Z g(x, y) = 0
L(x, y, λ) := f(x, y) + λg(x, y) = sin x + sin y + λ(cos x − cos y + 1).
∂xL(x, y, λ) = cos x − λ sin x
∂yL(x, y, λ) = cos y + λ sin y.
∂xL(x, y) = ∂yL(x, y) = 0 sin x = 0 sin y = 0 cot x = cot(−y)
x = −y + kπ k ∈ Z cos x − cos(−x + kπ) + 1 = 0
cos x − cos(x − kπ) + 1 = 0 cos x − (−1) k cos x + 1 = 0 (1 − (−1) k ) cos x = −1
k = 2j k = 2j + 1 cos x = −1/2
x1 = 2/3π + 2hπ x2 = 4/3π + 2hπ h ∈ Z y1 = −2/3π + 2hπ + (2j + 1)π =
π/3 + 2(j − h)π y2 = −4/3π − 2hπ + (2j + 1)π = −1/3π + 2(j − h)π
m = j − h ∈ Z
Phm =
2 π
4
π + 2hπ, + 2mπ , Qhm = π + 2hπ, −2 π + 2mπ .
3 3 3 3
h, m ∈ Z f(Phm) = √ 3/2 + √ 3/2 = √ 3 f(Qhm) =
− √ 3/2 − √ 3/2 = − √ 3
f(x, y) = 3x 2 +4y 2 −6x+3
x 2 + y 2 = 4
g(x, y) = x 2 + y 2 − 4 ∇g(x, y) = (0, 0) x = y = 0
(0, 0) g(x, y) = 0 g(0, 0) = 0
L(x, y, λ) := f(x, y) + λg(x, y) = 3x 2 + 4y 2 − 6x + 3 + λ(x 2 + y 2 − 4).
∂xL(x, y, λ) = 6x − 6 − 2λx
∂yL(x, y, λ) = 8y − 2λy = 2y(4 − λ).
∂xL(x, y) = ∂yL(x, y) = 0 y = 0 λ = 4
y = 0 x = ±2 λ = 4
x = −3 9 + y 2 = 4
(2, 0) (−2, 0) f(2, 0) = 3 f(−2, 0) = 27
g
x = 2 cos θ y = 2 sin θ θ ∈ [0, 2π[
h(θ) = f(2 cos θ, 2 sin θ) = 12 cos 2 θ + 16 sin 2 θ − 12 cos θ + 3 = 4 sin 2 θ − 12 cos θ + 15.
h ′ (θ) = 8 sin θ cos θ + 12 sin θ = 4 sin θ(2 cos θ + 3),
h ′′ (θ) = 4 cos θ(2 cos θ + 3) + 4 sin θ(−2 sin θ).
h ′ (θ) = 0 θ = 0, π h ′′ (0) = 20 > 0 h ′′ (π) = −4 < 0 (2, 0) (−2, 0)
f(x, y) = x(x 2 + y 2 )
xy = 1