The document discusses solving an optimization problem to find the maximum value of a function w subject to constraints. It is determined that the fixed point wF is 3c and the sliding point wS is 48/13c, which is greater than the fixed point. Tables are presented showing the variables and values at the fixed and sliding points.
4. ղ
λεΫͷޭ֬࿑ಇऀ͕ྗͨ͠߹ Ͱ ྗ͠ͳ͍߹ Ͱ͋Δͱ͢Δ
ྗͨ͠߹ʹ ͷίετ͕࿑ಇऀʹ͔͔Δͱ͢Δ
λεΫ͕ޭͨ͠ͱ͖ ༻ޏओ ͷརӹΛಘͯ ࿑ಇऀۚ ΛಘΔࣦഊͷ߹ʹ͍ޓԿಘͳ͍ͱ͢Δ
༻ޏओ͓Αͼ࿑ಇऀʹ͍ͭͯϦεΫதཱతͳޮ༻ؔ ΛͭͱԾఆ͢Δ
ྗ͢Δ͜ͱΛ͖Ͱ੍ڧΔ߹ͷ࠷దͳۚ ΛٻΊΑࢀՃ੍ͷΈߟ͑ͨ࠷దԽΛղ͚ྑ͍
TU
੍݅ ΑΓ
త͕ؔ࠷େͱͳΔͷ
Ώ͑ʹ࠷దͳۚ ͱͳΔ
1
3
1
16
c 0
R 0 w
u(x) = x
wF
max
w
1
3
(R − w)
1
3
(w − c) +
2
3
(−c) ≥ 0
1
3
(w − c) +
2
3
(−c) ≥ 0 ⇔ w ≥ 3c
w = 3c
wF
= 3c
w ࢀՃ
ڋ൱
ྗ͢Δ
ྗ͠ͳ͍
ޭ
ࣦഊ
ޭ
ࣦഊ
༻ޏओ ࿑ಇऀ
ࣗવ
R − w w − c
D
ڋ൱
ڋ൱
5. ղ
λεΫͷޭ֬࿑ಇऀ͕ྗͨ͠߹ Ͱ ྗ͠ͳ͍߹ Ͱ͋Δͱ͢Δ
ྗͨ͠߹ʹ ͷίετ͕࿑ಇऀʹ͔͔Δͱ͢Δ
λεΫ͕ޭͨ͠ͱ͖ ༻ޏओ ͷརӹΛಘͯ ࿑ಇऀۚ ΛಘΔࣦഊͷ߹ʹ͍ޓԿಘͳ͍ͱ͢Δ
༻ޏओ͓Αͼ࿑ಇऀʹ͍ͭͯϦεΫதཱతͳޮ༻ؔ ΛͭͱԾఆ͢Δ
ྗ͢Δ͜ͱΛ͍ͳ͖Ͱ੍ڧ߹ͷ࠷దͳۚ ΛٻΊΑ
TU
੍݅ ͔ͭ
త͕ؔ࠷େͱͳΔͷ
Ώ͑ʹ࠷దͳۚ ͱͳΔ
1
3
1
16
c 0
R 0 w
u(x) = x
wS
max
w
1
3
(R − w)
1
3
(w − c) +
2
3
(−c) ≥ 0
1
3
(w − c) +
2
3
(−c) ≥
1
16
w +
15
16
0
1
3
(w − c) +
2
3
(−c) ≥ 0 ⇔ w ≥ 3c
1
3
(w − c) +
2
3
(−c) ≥
1
16
w +
15
16
0 ⇔ w ≥
48
13
c
w =
48
13
c
wS
=
48
13
c( 3c = wF
)
w ࢀՃ
ڋ൱
ྗ͢Δ
ྗ͠ͳ͍
ޭ
ࣦഊ
ޭ
ࣦഊ
༻ޏओ ࿑ಇऀ
ࣗવ
R − w w − c
D
R − w w
༠Ҿཱ྆ੑ݅
༠Ҿཱ྆ੑ݅ͷͱͰ
ྗ͢ΔΛબͤΔ࣌ͷظརಘ
6.
༻ޏओ ਓͷ࿑ಇऀ͘͠ਓͷ࿑ಇऀͱܖΛ͠ ͋ΔλεΫͷ࣮ߦΛߟ͍͑ͯΔ
ਓͷ࿑ಇऀΛͨͬޏ߹ʹ͍ͭͯ
λεΫͷޭ֬ ӈදͷΑ͏ʹͳΔͱ͢Δ
ྗͨ͠߹ʹ ͷίετ͕࿑ಇऀʹ͔͔Δͱ͢Δ
λεΫ͕ޭͨ͠ͱ͖ ༻ޏओ ͷརӹΛಘͯ ਓͷ࿑ಇऀͦΕͧΕۚ ΛಘΔ
ࣦഊͷ߹શһԿಘͳ͍ͱ͢Δ·ͨ ͦͦܖΛ݁ͳ͍߹ʹ͍ޓԿಘͳ͍ͱ͢Δ
༻ޏओ͓Αͼ࿑ಇऀʹ͍ͭͯϦεΫதཱతͳޮ༻ؔ ΛͭͱԾఆ͢Δ
ਓͷ࿑ಇऀʹྗ͢Δ͜ͱΛ͖Ͱ੍ڧΔ߹ʹ࠷దͳۚ ΛٻΊΑ
ࢀՃ੍Λߟ͑ͯ ࠷దԽΛղ͚ྑ͍
ਓͷ࿑ಇऀʹྗ͢Δ͜ͱΛ͍ͳ͖Ͱ੍ڧ߹ʹ࠷దͳۚ ΛٻΊΑ
ࢀՃ੍͓Αͼ༠Ҿཱ྆ੑ݅ ࿑ಇऀʹͱͬͯྗ͢Δํ͕ ྗ͠ͳ͍ΑΓಘ͢ΔͨΊͷ݅
Λߟ͑ͯ ࠷దԽΛղ͚ྑ͍ˎྗͤͨ͞ํ͕༻ޏओʹͱͬͯಘͰ͋Δͱͯ͠ྑ͍
c 0
R 0 w′

u(x) = x
w′

F
w′

S
࿑ಇऀ ྗ͠ͳ͍ ྗ͢Δ
ྗ͠ͳ͍
ྗ͢Δ
7.
࿑ಇऀ ྗ͠ͳ͍ ྗ͢Δ
ྗ͠ͳ͍
ྗ͢Δ
w′

ࢀՃ
ڋ൱
ྗ͢Δ
ྗ͠ͳ͍
༻ޏओ ࿑ಇऀ
ࣗવ
R − 2w′

w′

− c w′

− c
0 −c −c
R − 2w′

w′

− c w′

࿑ಇऀ
R − 2w′

w′

w′

− c
0 0 −c
0 0 0
R − 2w′

w′

w′

ྗ͢Δ
ྗ͠ͳ͍
ྗ͢Δ
ྗ͠ͳ͍
0 −c 0
8. ղ
ਓͷ࿑ಇऀʹྗ͢Δ͜ͱΛ͖Ͱ੍ڧΔ߹ʹ࠷దͳۚ ΛٻΊΑ
ࢀՃ੍Λߟ͑ͯ ࠷దԽΛղ͚ྑ͍
TU
త͕ؔ࠷େͱͳΔͷ
w′

F
max
w′

2
3
(R − 2w′

)
2
3
(w′

− c) +
1
3
(−c) ≥ 0
w′

F
=
3
2
c
࿑ಇऀ ྗ͠ͳ͍ ྗ͢Δ
ྗ͠ͳ͍
ྗ͢Δ
w′

ࢀՃ
ڋ൱
ྗ͢Δ
ྗ͠ͳ͍
༻ޏओ ࿑ಇऀ
ࣗવ
R − 2w′

w′

− c w′

− c
0 −c −c
R − 2w′

w′

w′

࿑ಇऀ
R − 2w′

w′

w′

0 0 −c
0 0 0
R − 2w′

w′

w′

ྗ͢Δ
ྗ͠ͳ͍
ྗ͢Δ
ྗ͠ͳ͍
0 −c 0
9. ਓͷ࿑ಇऀʹྗ͢Δ͜ͱΛ͍ͳ͖Ͱ੍ڧ߹ʹ࠷దͳۚ ΛٻΊΑ
ˎਓͷ࿑ಇऀʹྗͤͨ͞ํ͕༻ޏओʹͱͬͯಘͰ͋Δͱͯ͠ྑ͍
૬खͷͯ͢ߦಈʹରͯ͠ ʮྗ͢Δʯ͕࠷దͱͳΔ݅Λߟ͑Δ
࿑ಇऀͷʮྗ͠ͳ͍ʯʹର͠ ྗ͢Δͷ͕࠷దʹͳΔͷ
࿑ಇऀͷʮྗ͢Δʯʹର͠ ྗ͢Δͷ͕࠷దʹͳΔͷ
Ώ͑ʹ ͷ͕݅ྗ͢ΔͨΊͷ݅Ͱ͋Δ
w′

S
1
8
w′

≤
1
3
w′

− c ⇔ w′

≥
24
5
c
1
3
w′

≤
2
3
w′

− c ⇔ w′

≥ 3c
w′

≥
24
5
c
ղ
࿑ಇऀ ྗ͠ͳ͍ ྗ͢Δ
ྗ͠ͳ͍
ྗ͢Δ
w ࢀՃ
ڋ൱
ྗ͢Δ
ྗ͠ͳ͍
༻ޏओ ࿑ಇऀ
ࣗવ
R − 2w′

w′

− c w′

− c
0 −c −c
R − 2w′

w′

− c w′

࿑ಇऀ
R − 2w′

w′

w′

− c
0 0 −c
0 0 0
R − 2w′

w′

w′

ྗ͢Δ
ྗ͠ͳ͍
ྗ͢Δ
ྗ͠ͳ͍
0 −c 0
࿑ಇऀ ྗ͠ͳ͍ ྗ͢Δ
ྗ͠ͳ͍ X` X`
ྗ͢Δ X`D X`D
͍͑ݴΕ ʮྗ͢Δʯ͕
ऑ ࢧઓུͱͳΔͨΊͷ݅
10. ղ
ਓͷ࿑ಇऀʹྗ͢Δ͜ͱΛ͍ͳ͖Ͱ੍ڧ߹ʹ࠷దͳۚ ΛٻΊΑ
ˎਓͷ࿑ಇऀʹྗͤͨ͞ํ͕༻ޏओʹͱͬͯಘͰ͋Δͱͯ͠ྑ͍
TU
త͕ؔ࠷େͱͳΔͷ
w′

S
max
w′

2
3
(R − 2w′

)
2
3
(w′

− c) +
1
3
(−c) ≥ 0 ⇔ w′

≥
3
2
c
w′

≥
24
5
c
w′

S
=
24
5
c
࿑ಇऀ ྗ͠ͳ͍ ྗ͢Δ
ྗ͠ͳ͍
ྗ͢Δ
ࢀՃ
ڋ൱
ྗ͢Δ
ྗ͠ͳ͍
࿑ಇऀ
ࣗવ
R − 2w′

w′

− c w′

− c
0 −c −c
R − 2w′

w′

− c w′

࿑ಇऀ
R − 2w′

w′

w′

− c
0 0 −c
0 0 0
R − 2w′

w′

w′

ྗ͢Δ
ྗ͠ͳ͍
ྗ͢Δ
ྗ͠ͳ͍
0 −c 0
w′

3
2
c =
15
10
c 3c =
30
10
c
24
5
c =
48
10
c
૬ख͕ྗ͠ͳͯ͘ྗ͢Δ
૬ख͕ྗ͠ͳ͍ͷͰ͋Ε
ྗ͠ͳ͍
ࢀՃ੍