1. The document discusses maximizing a function w subject to constraints. It derives that the maximum value wF is equal to (3c)2 and the stationary value wS is equal to (48/13)c2. 2. It also considers the boundary conditions for the functions u(x)=x at x=0 and x=R. 3. The document concludes that the maximum occurs at the boundary condition rather than the stationary point, and discusses implications for the original problem.

1. ήʔϜཧ࿦#4*$ԋश
࠷ద௞ۚઃఆ

2. ໰୊
ղ౴
ߟ࡯

3. ໰୊
‫༻ޏ‬ओ͸ ਓͷ࿑ಇऀͱ‫ܖ‬໿Λ͠ ͋ΔλεΫͷ࣮ߦΛߟ͍͑ͯΔ
λεΫͷ੒ޭ֬཰͸ ࿑ಇऀ͕౒ྗͨ͠৔߹͸ Ͱ ౒ྗ͠ͳ͍৔߹͸ Ͱ͋Δͱ͢Δ
౒ྗͨ͠৔߹ʹ͸ ͷίετ͕࿑ಇऀʹ͔͔Δͱ͢Δ
λεΫ͕੒ޭͨ͠ͱ͖ ‫༻ޏ‬ओ͸ ͷརӹΛಘͯ ࿑ಇऀ͸௞ۚ ΛಘΔࣦഊͷ৔߹͸‫ʹ͍ޓ‬Կ΋ಘͳ͍ͱ͢Δ
·ͨ ͦ΋ͦ΋‫ܖ‬໿Λ݁͹ͳ͍৔߹΋‫ʹ͍ޓ‬Կ΋ಘͳ͍ͱ͢Δ
‫༻ޏ‬ओ͸ಘΒΕΔརӹʹରͯ͠ ϦεΫதཱతͳޮ༻ؔ਺ Λ΋ͪ
࿑ಇऀ͸ใुʹରͯ͠ ϦεΫճආతͳޮ༻ؔ਺ Λ΋ͭͱԾఆ͢Δ
DG#4*$ԋशͰ͸ͲͪΒ΋ϦεΫதཱతͳޮ༻ؔ਺Λ΋ͭͱԾఆͨ͠
౒ྗ͢Δ͜ͱΛ‫͖Ͱ੍ڧ‬Δ৔߹ͷ࠷దͳ௞ۚ Λ‫ٻ‬ΊΑ
ࢀՃ੍໿ͷΈߟ͑ͨ࠷దԽ໰୊Λղ͚͹ྑ͍
౒ྗ͢Δ͜ͱΛ‫͍ͳ͖Ͱ੍ڧ‬৔߹ͷ࠷దͳ௞ۚ Λ‫ٻ‬ΊΑ
ࢀՃ੍໿͓Αͼ༠Ҿཱ྆ੑ৚݅ ࿑ಇऀʹͱͬͯ౒ྗ͢Δํ͕ ౒ྗ͠ͳ͍ΑΓ΋ಘ͢ΔͨΊͷ৚݅
Λߟ͑ͯ ࠷దԽ໰୊Λղ͚͹ྑ͍ˎ౒ྗͤͨ͞ํ͕‫༻ޏ‬ओʹͱͬͯಘͰ͋Δͱͯ͠ྑ͍
#4*$ԋशࢀর
1
3
1
16
c 0
R 0 w
u(x) = x
u(x) = x
wF
wS
͸े෼େ͖ͳ஋
R 0

4. ղ౴
λεΫͷ੒ޭ֬཰͸ ࿑ಇऀ͕౒ྗͨ͠৔߹͸ Ͱ ౒ྗ͠ͳ͍৔߹͸ Ͱ͋Δͱ͢Δ
౒ྗͨ͠৔߹ʹ͸ ͷίετ͕࿑ಇऀʹ͔͔Δͱ͢Δ
λεΫ͕੒ޭͨ͠ͱ͖ ‫༻ޏ‬ओ͸ ͷརӹΛಘͯ ࿑ಇऀ͸௞ۚ ΛಘΔࣦഊͷ৔߹͸‫ʹ͍ޓ‬Կ΋ಘͳ͍ͱ͢Δ
·ͨ ͦ΋ͦ΋‫ܖ‬໿Λ݁͹ͳ͍৔߹΋‫ʹ͍ޓ‬Կ΋ಘͳ͍ͱ͢Δ
‫༻ޏ‬ओ͸ಘΒΕΔརӹʹରͯ͠ ϦεΫதཱతͳޮ༻ؔ਺ Λ΋ͪ
࿑ಇऀ͸ใुʹରͯ͠ ϦεΫճආతͳޮ༻ؔ਺ Λ΋ͭͱԾఆ͢Δ
౒ྗ͢Δ͜ͱΛ‫͖Ͱ੍ڧ‬Δ৔߹ͷ࠷దͳ௞ۚ Λ‫ٻ‬ΊΑࢀՃ੍໿ͷΈߟ͑ͨ࠷దԽ໰୊Λղ͚͹ྑ͍
TU
੍໿৚݅ ΑΓ
໨తؔ਺͕࠷େͱͳΔͷ͸
Ώ͑ʹ࠷దͳ௞ۚ ͱͳΔ
1
3
1
16
c 0
R 0 w
u(x) = x
u(x) = x
wF
max
w
1
3
(R − w)
1
3
( w − c) +
2
3
(0 − c) ≥ 0
1
3
( w − c) +
2
3
(0 − c) ≥ 0 ⇒ w ≥ 9c2
w = 9c2
wF
= 9c2
w ࢀՃ
‫ڋ‬൱
౒ྗ͢Δ
౒ྗ͠ͳ͍
੒ޭ
ࣦഊ
੒ޭ
ࣦഊ
‫༻ޏ‬ओ ࿑ಇऀ
ࣗવ
R − w w − c
D
‫ڋ‬൱
‫ڋ‬൱

5. ղ౴
λεΫͷ੒ޭ֬཰͸࿑ಇऀ͕౒ྗͨ͠৔߹͸ Ͱ ౒ྗ͠ͳ͍৔߹͸ Ͱ͋Δͱ͢Δ
౒ྗͨ͠৔߹ʹ͸ ͷίετ͕࿑ಇऀʹ͔͔Δͱ͢Δ
λεΫ͕੒ޭͨ͠ͱ͖ ‫༻ޏ‬ओ͸ ͷརӹΛಘͯ ࿑ಇऀ͸௞ۚ ΛಘΔࣦഊͷ৔߹͸‫ʹ͍ޓ‬Կ΋ಘͳ͍ͱ͢Δ
‫༻ޏ‬ओ͸ಘΒΕΔརӹʹରͯ͠ ϦεΫதཱతͳޮ༻ؔ਺ Λ΋ͪ ࿑ಇऀ͸ใुʹରͯ͠ ϦεΫճආతͳޮ༻ؔ਺ Λ΋ͭͱԾఆ͢Δ
౒ྗ͢Δ͜ͱΛ‫͍ͳ͖Ͱ੍ڧ‬৔߹ͷ࠷దͳ௞ۚ Λ‫ٻ‬ΊΑ
TU
੍໿৚݅ ͔ͭ
໨తؔ਺͕࠷େͱͳΔͷ͸
Ώ͑ʹ࠷దͳ௞ۚ ͱͳΔ
1
3
1
16
c 0
R 0 w
u(x) = x 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
(0 − c) ≥ 0 ⇒ w ≥ (3c)2
1
3
( w − c) +
2
3
(−c) ≥
1
16
w +
15
16
0 ⇒ w ≥
(
48
13
c
)
2
w =
(
48
13
c
)
2
wS
=
(
48
13
c
)
2
( (3c)2
= wF
)
w ࢀՃ
‫ڋ‬൱
౒ྗ͢Δ
౒ྗ͠ͳ͍
੒ޭ
ࣦഊ
੒ޭ
ࣦഊ
‫༻ޏ‬ओ ࿑ಇऀ
ࣗવ
R − w w − c
D
R − w w
༠Ҿཱ྆ੑ৚݅
༠Ҿཱ྆ੑ৚݅ͷ΋ͱͰ
౒ྗ͢ΔΛબ͹ͤΔ࣌ͷ‫ظ‬଴རಘ

6. ߟ࡯
࿑ಇऀ͕ϦεΫճආత ‫༻ޏ‬ओ͕ϦεΫதཱతͷ৔߹
࿑ಇऀͷ‫ظ‬଴ޮ༻
ϑΝʔετϕετͷ৔߹
ηΧϯυϕετͷ৔߹
‫༻ޏ‬ओͷ‫ظ‬଴ޮ༻
ϑΝʔετϕετͷ৔߹
ϑΝʔετϕετͷ৔߹
wF
= (3c)2
wS
=
(
48
13
c
)
2
1
3
( (3c)2
− c) +
2
3
(−c) = 0
1
3
(
(
48
13
c
)
2
− c) +
2
3
(−c) =
3
13
c
1
3
(R − (3c)2
)
1
3(
R−
(
48
13
c
)
2
)
ͲͪΒ΋ϦεΫதཱతͳ৔߹DG#4*$ԋश
࿑ಇऀͷ‫ظ‬଴ޮ༻
ϑΝʔετϕετͷ৔߹
ηΧϯυϕετͷ৔߹
‫༻ޏ‬ओͷ‫ظ‬଴ޮ༻
ϑΝʔετϕετͷ৔߹
ϑΝʔετϕετͷ৔߹
wF
= 3c wS
=
48
13
c
1
3
(3c − c) +
2
3
(−c) = 0
1
3
(
48
13
c − c) +
2
3
(−c) =
3
13
c
1
3
(R − 3c)
1
3(
R −
48
13
c
)
͕े෼େ͖͍ͱ͢Ε͹ ‫༻ޏ‬ओͷෛ୲͕େ͖͘ͳΔ
ϦεΫΛऔΓͨ͘ͳ͍࿑ಇऀʹ ౒ྗΠϯηϯςΟϒΛ༩͑ΔͨΊʹϦεΫΛෛΘͤΔඞཁ͕͋Γ
ϦεΫΛෛΘͤΔ෼͚ͩ௥Ճඅ༻͕͔͔Δ
c

7. ήʔϜཧ࿦#4*$ԋश
࠷ద௞ۚઃఆ