極値, 最大最小問題, ラグランジュの未定乗数法 - 名古屋大学

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K111
, ,
: November 24, 2008
.
, .
, .
Step .
z = f(x, y) f(x, y) .
f(x, y) , .
z = f(x, y) .
(1) f(x, y) = x 2 + y 2 (2) f(x, y) = x 2 − y 2 (3) f(x, y) = x 2
.
z = x 2 + y 2 . z = x 2 − y 2
, .
. z = x 2 x = 0
, . (.)
.
.
1 a, b, c , f(x, y) = ax 2 +bxy +cy 2 . D = 4ac−b 2
f(x, y) . (a, b, c 0 .)
α, β, γ, δ X = αx + βy, Y = γx + δy
(1) D > 0 . f(x, y) a > 0 X 2 + Y 2 , a < 0 −X 2 − Y 2
.
(2) D < 0 . f(x, y) X 2 − Y 2 .
(3) D = 0 . f(x, y) ±X 2 .
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y = f(x) ,
. f ′ (α) = 0 α . f ′′ (α)
. f ′′ (α) ≠ 0 , f ′′ (α) = 0 ,
, . .
f ′ f ′′ .
f ′ (f x , f y ) , f ′′ .
C 2 f(x, y) z = f(x, y) . (a, b)
, xy-
, f x (a, b) = f y (a, b) = 0. (a, b)
, (x, y) = (a, b)
f(x, y) − f(a, b) = AX 2 + BXY + CY 2 + o(X 2 + Y 2 ). (1)
A = f xx (a, b)/2, B = f xy (a, b), C = f yy (a, b)/2 , X = x − a, Y = y − b
() . o(X 2 + Y 2 ) (a, b)
AX 2 + BXY + CY 2
. 4AC − B 2 ,
. (.)
2 z = f(x, y)
(1) f x (a, b) = f y (a, b) = 0 (a, b)
(2) (a, b)
(
)
f xx (a, b) f xy (a, b)
H(a, b) =
f yx (a, b) f yy (a, b)
det H(a, b)
(a) det H(a, b) > 0 f xx (a, b) < 0
(b) det H(a, b) > 0 f xx (a, b) > 0
(c) det H(a, b) < 0
(d) det H(a, b) = 0
1. i) Z = AX 2 + BXY + CY 2 4AC − B 2 .
.
ii) det H(a, b) = 0 . (1)
o(X 2 + Y 2 ) . , det H(a, b) = 0
4AC − B 2 = 0 , z = f(x, y) x 2
. f(x, y) x 2 + y 4 , x 2 − y 4 , x 2 + y 3 o(x 2 + y 2 )
, () . (y 4 , −y 4 , y 3
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! .) 0 x 2 + y 2 x 2 − y 2
o(x 2 + y 2 ) .
2. , .
, .
, .
0 , .
0 . ,
I §8 . 8.3 8.4 .
.
x, y g(x, y) = 0 , z = f(x, y)
. .
, x 2 + y 2 ≤ 1 z = f(x, y)
, x 2 + y 2 < 1 ,
. ,
x 2 + y 2 = 1 z = f(x, y) .
.
g(x, y) = 0 f(x, y) (a, b)
, (g x (a, b), g y (a, b)) ≠ 0
(f x (a, b), f y (a, b)) = λ (g x (a, b), g y (a, b))
λ
g(x, y) = 0, f x (a, b) = λ g x (a, b), f y (a, b) = λ g y (a, b) (2)
a, b, λ ,
a, b, λ () . (
(a, b) .)
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3. i) (2) F (x, y, λ) = f(x, y) − λg(x, y) ()
F x (x, y, λ) = F y (x, y, λ) = F λ (x, y, λ) = 0
. F (x, y, λ) () ,
.
ii) ,
, . ,
.
3 F (x, y, λ) i) . (x, y, λ) = (a, b, λ 0 ) F (x, y, λ)
. (x, y, λ) = (a, b, λ 0 ) F (x, y, λ)
, .
(bordered hessian) , .
∣
0 g x (a, b) g y (a, b)
g x (a, b) F xx (a, b, λ 0 ) F xy (a, b, λ 0 )
g y (a, b) F yx (a, b, λ 0 ) F yy (a, b, λ 0 )
(0 .)
,
, . (.)
iii) (a, b) , g(x, y) = 0
(g x (a, b), g y (a, b)) ≠ 0 . (g x (a, b), g y (a, b)) = 0 , (a, b)
g(x, y) = g x (x, y) = g y (x, y) = 0 ,
(a, b) .
, .
iv) . k
f(x, y) = k x, y () . k
. k = k 1 , k 2 , k 3 .
g(x, y) = 0
f(x, y) = k 3
f(x, y) = k 2
f(x, y) = k 1
(a,b)
−→ n
∣
l
k = k 1 f(x, y) = k 1 g(x, y) = 0 , g(x, y) = 0
f(x, y) = k 1 (x, y) . k 1 . (f(x, y)
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k 1 .) k = k 3 g(x, y) = 0 (x, y) f(x, y) = k 3
. k 3 ɛ > 0 f(x, y) = k 3 ± ɛ()
, g(x, y) = 0 , f(x, y) = k 3 ± ɛ (x, y) .
k 3 . f(x, y) = k 2
g(x, y) = 0 . (a, b)
g(x, y) = 0 f(x, y) = k 2 l() . −→ n
. K109 , g(x, y) = 0
(a, b) (g x (a, b), g y (a, b)) , f(x, y) = k (a, b)
(f x (a, b), f y (a, b)) .
, (f x (a, b), f y (a, b)) = λ(g x (a, b), g y (a, b)) λ
.
,
. K109 f = f(x 1 , . . . , x n )
, gradf grad f = (f x1 , . . . , f xn ) .
()
g i = g i (x 1 , . . . , x n ) = 0 (i = 1, . . . , m)
f = f(x 1 , . . . , x n ) a = (a 1 , . . . , a n ) .
, g i ( ∂g i
∂x j
) i,j m , grad f(a) (grad g i (a)) i
(m ) . λ i (i = 1, . . . , m)
.
m∑
grad f(a) = λ i grad g i (a). (3)
i=1
4. i) (3) g i = 0 n + m ,
x 1 , . . . , x n , λ 1 , . . . , λ m n + m , () .
ii) (3) n + m () .
F (x 1 , . . . , x n , λ 1 , . . . , λ m ) = f(x 1 , . . . , x n ) −
m∑
λ i g i (x 1 , . . . , x n ).
i=1
iii) , ,
.
. ,
. .
, .
, , A.C. , (),
. ,
.
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y = f(x) [a, b]
. [a, b] (a, b) y = f(x)
.
, y = f(x) . f(a), f(b)
(a, b) .
.
f(x, y) D , D
. , f(x, y)
. D f(x, y) .
, .
.
D R n . D
f(x 1 , . . . , x n ) D .
. x 2 + y 2 ≤ 1
. x 2 + y 2 < 1 .
.
, R n
. ,
. , .
, (D ) ,
.
.
, .
.
(a, b) , ,
. . (.)
, !
(Examples)
1.
.
f(x, y) = sin x + sin y + sin(x + y) (0 < x < 2π, 0 < y < 2π)
(f x = f y = 0 ) .
f x (x, y) = cos x + cos(x + y) = 0, f y (x, y) = cos y + cos(x + y) = 0
, cos x = cos y. x, y 0 < x < 2π, 0 < y < 2π ,
x = y x + y = 2π . x = y f x = 0
f x = cos x + cos(2x) = cos x + 2 cos 2 x − 1 = (2 cos x − 1)(cos x + 1) = 0
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, x = π cos x = 1/2. (x, y) = (π, π), (π/3, π/3),
(5π/3, 5π/3) . x + y = 2π (x, y) = (π, π)
.
.
H(x, y) =
(
f xx (x, y) f xy (x, y)
f yx (x, y) f yy (x, y)
)
=
(
− sin x − sin(x + y) − sin(x + y)
− sin(x + y) − sin y − sin(x + y)
det H(π, π) = 0, det H(π/3, π/3) = det H(5π/3, 5π/3) = 9/4 > 0.
(x, y) = (π/3, π/3) f(x, y) 3 √ 3/2 , (x, y) = (5π/3, 5π/3)
f(x, y) −3 √ 3/2 . (x, y) = (π, π)
. f(x, x) = 2 sin x + sin(2x) = 2 sin x(1 + cos x) , x π ,
x < π f(x, x) > 0 , x > π f(x, x) < 0 . (x, y) = (π, π)
f(x, y) f(π, π) = 0 .
(x, y) = (π, π) .
)
.
2.
, .
x, y, z .
sin x+sin y+sin z .
π = x+y+z ,
sin x + sin y + sin(π − x − y) = sin x + sin y + sin(x + y)
x, y > 0 x + y < π x, y
. x = y = π/3
3 √ 3/2 .
, .
,
. .
sin x + sin y + sin(x + y) x, y ≥ 0 x + y ≤ π . (
, . )
x, y ≥ 0 x + y ≤ π
. . (x = 0
) 2 , (x, y) = (π/3, π/3)
3 √ 3/2 . .
, (x, y) = (π/3, π/3) .
(x, y) = (π/3, π/3) 3 √ 3/2 .
.
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3. x 2 + y 2 = 1 f(x, y) = x 3 + y 3 − 3x − 3y
x 2 + y 2 = 1
. . y = ± √ 1 − x 2 f(x, y) , x
.
. . g(x, y) = x 2 + y 2 − 1 .
g = g x = g y = 0
.
F (x, y, λ) = f(x, y) − λg(x, y)
.
F x (x, y, λ) = 3x 2 −3−2λx = 0, F y (x, y, λ) = 3y 2 −3−2λy = 0, F λ (x, y, λ) = −x 2 −y 2 +1 = 0
. x, y 3t 2 − 2λt − 3 = 0 . x ≠ y
x, y .
x 2 + y 2 = (x + y) 2 − 2xy = 4λ 2 /9 + 2 = 1 . λ
. x = y . x = y = √ 2/2
x = y = − √ 2/2 , F λ .
, .
, , .
(x, y) = (− √ 2/2, − √ 2/2) , 5 √ 2/2
. (x, y) = ( √ 2/2, √ 2/2) ,
−5 √ 2/2 .
4. a 1 , . . . , a n 0 . n R n
, H : a 1 x 1 +· · ·+a n x n = b |b|/ √ a 2 1 + · · · + a2 n
. (.)
a 1 x 1 +· · ·+a n x n = b √ x 2 1 + · · · + x2 n
. x 2 1 +· · ·+x2 n
. . f(x 1 , . . . , x n ) = x 2 1 +· · ·+x2 n ,
g(x 1 , . . . , x n ) = a 1 x 1 + · · · + a n x n − b ,
a 1 x 1 + · · · + a n x n = b, 2x 1 − λa 1 = 0, . . . , 2x n − λa n = 0
. x i = λa i /2 , λ = 2b/(a 2 1 + · · · + a2 n ) . z i =
ba i /(a 2 1 + · · · + a2 n ) , (z 1 , . . . , z n ) .
f(z 1 , . . . , z n ) =
b 2
a 2 1 + · · · + a 2 n
. .
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.
R |b|/ √ a 2 1 + · · · + a2 n . H |x i | ≤ R (i = 1, . . . , n)
D . D (z 1 , . . . , z n ) . D
, .
H . (y 1 , . . . , y n ) , i y i > R
,
f(y 1 , . . . , y n ) > b 2 /(a 2 1 + · · · + a 2 n) = f(z 1 , . . . , z n )
. f(y 1 , . . . , y n ) . f(x 1 , . . . , x n )
H . , f(x 1 , . . . , x n )
. (z 1 , . . . , z n )
, .
(Problems)
1. i) f(x, y) = x 2 − xy + 2y 2 − x − 2y .
ii) f(x, y) = (x + y)e −xy .
2. .
3. x > 0, y > 0, z > 0 .
.
i) x 2 + y 2 + z 2 = a 2 , xy + yz + zx.
ii) xy + yz + zx = k 2 , xyz.
4. (2 ) . C 1
f(x, y), g(x, y) , g(x, y) = 0 f(x, y) (a, b)
, (g x (a, b), g y (a, b)) ≠ 0
λ .
(f x (a, b), f y (a, b)) = λ (g x (a, b), g y (a, b))
5. i) . (.)
ii) . (.)
6. n A i .
P A 2 1 + · · · + P A2 n
, .
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7. p 1 , . . . , p n > 0, p 1 + · · · + p n = 1 ,
n∑
H := − p k log p k
k=1
, .
8. y 2 = x 3 . y 2 = x 3 , f(x, y) =
y − (x + 1) 2 . (?)
9. . : y = f(x) , f ′ (α) = 0
x = α 2 f ′′ (α) .
.
10. i) y = f(x) (a, b) C 1 . f ′ (α) = 0
, f(x) , .
ii) z = f(x, y) R 2 . f (f x = f y = 0
) , z = f(x, y) . z = f(x, y)
(No! .)
11. , ,
. (.)
12. , ,
. (.)
(Homework)
13. i) f(x, y) = x 3 + y 3 − x 2 + xy − y 2 .
ii) f(x, y) = xy(a − x − y) .
14. ,
.
15. 2xy + z 2 − 1 = 0 (1, 1, 2) . (
.)
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