pdf (it, 2.898,97 KB, 25/01/13)

⎧⎪
2x + λ(2x + y) + 2µx = 0
⎨2y
+ λ(2y + x) + 2µy = 0
2z − 2λz = 0
⎪⎩
x2 + y2 + xy − z2 = 1
x2 + y2 = 1
z 2 = xy z = 0 λ = 1
z = 0 xy = 0 x = 0 y = 0 x =
y = 0 x = 0 y = ±1 y = 0 x = ±1
(±1, 0, 0) (0, ±1, 0) λ = 1
2(x − y) + (x − y) + 2µ(x − y) = 0 (3 + 2µ)(x − y) = 0 x = y
µ = −3/2 x = y z = ±x x = y = ± √ 2/2
( √ 2/2, √ 2/2, ± √ 2/2) (− √ 2/2, − √ 2/2, ± √ 2/2) µ = −3/2 λ = 1
2(x + y) + 3(x + y) − 3x = 0 2x = −5y x, y
z 2 = xy
(±1, 0, 0) (0, ±1, 0) ( √ 2/2, √ 2/2, ± √ 2/2) (− √ 2/2, − √ 2/2, ± √ 2/2)
f(±1, 0, 0) = f(0, ±1, 0) = 1 f( √ 2/2, √ 2/2, ± √ 2/2) = f(− √ 2/2, − √ 2/2, ± √ 2/2) = 3/2
(±1, 0, 0) (0, ±1, 0)
f(x, y) = √ ye−x2−y2 x2 + (y − 1) 2 ≤ 1}
D := {(x, y) ∈ R 2 :
f D
f D f(x, y) ≥ 0 f(x, y) = 0 y = 0
D y = 0 O(0, 0)
f(x, y) = √ ye−x2e−y2 ≤ √ ye−y2 = f(0, y)
g(y) = √ ye−y2 0 ≤ y ≤ 2
g ′ (y) = e −y2
1
2 √
− 2y3/2 =
y e−y2
2 √ y (1 − 4y2 ),
]0, 2[ y = 1/2
g(0) = 0 < g(2) = √ 2e −4 √
2
< g(1/2) =
2 e−1/4 ,
y = 1/2 g [0, 2] (0, 1/2) f D
D f
∇f(x, y) =
−2x √ ye −x2 −y 2
, e−x2 −y 2
2 √ y − 2y3/2 e −x2 −y 2
D y > 0 x = 0 y = 1/2
f
⎛
⎞
Hess f(x, y) =
⎝ 4e−x2 −y2 x2√y − 2e−x2−y2√ y 4e−x2−y2 xy3/2 − e−x2 −y 2 √ x
y
4e−x2−y2 xy3/2 − e−x2 −y 2 √ x
y 4e−x2−y2 y5/2 − 4e−x2−y2√ y − e−x2 −y 2
4y3/2 Hess f(0, 1/2) =
√
2
− 4√ 0
e
0 − 2√2 4√
e
.
.
⎠ ,