1. §
0.06kg 0.04kg 0.07kg
0.08kg 0 0.04kg
321 ;; xxx
3,1,0 =≥ ixi .
307.0204.0106.0 xxx ++
304.02.0108.0 xxx ++
321 8.17.12 xxx ++
( ) ( )
( )
( ) 3.1,03
30004.0.008.0
50007.004.006.0
2
max8.17.121
321
321
321
=≥
≤++
≤++
→++=
jx
xxx
xxx
xxxxf
j
=
04.0008.0
07.004.006.0
A
=
300
500
B
( )321 ;; xxxx =
( )321 ;; xxxx =
( )321 ;; xxxx =
- 1 -
2. XN 1 2 3
3.5m 20h 4m 16h 3.8m 18h
2.8m 10h 2.6m 12h 2.5m 15h
-
-
-
-
321 ;; xxx
3,1,0 =≥ jx j
321 434035 xxx ++ ,
321 304245 xxx ++ ,
013210434035304245 321321321 ≥−+⇔++≥++ xxxxxxxxx
321 434035 xxx ++ ,
321
321
321
4.2382.2695.248
305.2426.2458.2
438.3404355.3
xxx
xxx
xxx
++
×+×+×
+
×+×+×
321
321
321
122411441150
301542124510
431840163520
xxx
xxx
xxx
++
×+×+×
+
×+×+×
321 xxx ++
- 2 -
3. ( ) ( )
( )
( ) 3,1,03
52000122411441150
100004.2382.2695.248
1500434035
013210
2
min1
321
321
321
321
321
=≥
≤++
≤++
≥++
≥−+
→++=
jx
xxx
xxx
xxx
xxx
xxxxf
j
=
−
=
52000
10000
1500
0
,
122411441150
4.2382.2695.248
434035
13210
BA
121, C2, C3
-
-
- kmT ×
( )3,2,1;2,1 == jixij 0≥∀→ ijji xCK
1
131211 xxx ++
2
232221 xxx ++
1
2111 xx +
2
- 3 -
4. 2212 xx +
3
2313 xx +
kmT × :
232221131211 634275 xxxxxx +++++
( ) ( )
( )
( ) ( )3,2,1;2,103
25
20
15
40
20
2
min6342751
2313
2212
2111
232221
131211
232221131211
==≥
=+
=+
=+
=++
=++
→+++++=
jix
xx
xx
xx
xxx
xxx
xxxxxxxf
ij
- 4 -
5. §
( ) ( ) ( )
( )
( ) ( ) ( ) ( ) { }nJJJJjyytuxJjxJjx
bxa
bxa
bxa
xcxf
jjj
n
j
ijij
n
j
ijij
n
j
ijij
n
j
jj
;;2;1;;;3
2
maxmin1
32132010
1
1
1
1
=∪∪∈′∈∈
≥
≤
=
→=
≤≥
=
=
=
=
∑
∑
∑
∑
-
Vector ( )nxxxx ;;; 21 =
-
-
( ) ( )
( )
( ) yytuxxxxx
xxxx
xxxx
xxx
xxxxx
xxxxxxf
′≤≥
≤+++
−≥++−
=+−
≤+++−
→+++−=
35241
4321
5321
321
54321
54321
;0;;0;3
1002
182
2024
1722
2
max5231
( ) ( ) ( )
( ) ( )
( ) ( )njx
mibxa
xcxf
j
n
j
ijij
n
j
jj
,13
,12
maxmin1
0
1
1
=
==
→=
≥
=
=
∑
∑
- 5 -
6. ( ) ( )
( )
( ) 5,1;03
172
18
032
2
min331
543
5432
4321
54321
=≥
=−+
−=+−−
=+−+
→+−+−=
jx
xxx
xxxx
xxxx
xxxxxxf
j
( ) ( ) ( )
( )
( )
( )
( )
( ) ( ) ( )
( )
( )
( )
=
=≥=
=++
=++
=++
→=
+
+
+
+
≥
++
++
++
=
∑
mnmm
nm
nm
nmm
ij
mnmnmmmm
nnmm
nnmm
n
j
jj
aa
aa
aa
A
xxxxx
mibnjx
bxaxax
bxaxax
bxaxax
xcxf
...1......00
...0......10
...0......01
,10;,13
........................................
2
maxmin1
1
212
111
121
0
11
221122
111111
1
( )mibi ,10 =≥ .
mxxx ;;; 21
( ) ( ) mibbbbxxxxx imnmm ,1,00;;0;;;;;;;;;; 21121 =∀≥=+
- 6 -
7. ( ) ( )
( )
( )
−−=
=≥
=+++
=+−+−
=++
→+−+−=
003121
104043
012002
6,1;03
2832
0443
2022
2
min331
654321
4321
6421
541
54321
A
xxxxxx
jx
xxxx
xxxx
xxx
xxxxxxf
j
563
( ) ( )0,20,0,28,0,0,,,, ,654,321 =xxxxxx
- 7 -
8. §
i
n
j
jij bxa ≤∑
=1
01 ≥+ix
in
n
j
jij bxxa =+ +
=
∑ 1
1
i
n
j
jij bxa ≥∑
=1
01 ≥+ix
in
n
j
jij bxxa =− +
=
∑ 1
1
0,0 ≥−=≤ jjjj ttxthaytax
0,, ≥′′′′′−′=′ jjjjjj xxxxxthaytayytux
( ) ( )
( )
( )
( )
( )
( )
( ) yytuxxxxx
dxxxx
cxxx
bxxx
axxxxx
xxxx
xxx
xxx
xxxxx
xxxxxxf
′≤≥
=+−+
≥++
≤−−−
≤+++−
⇔
=+−+
≥++
−≥++
≤+++−
→−++−=
32451
4321
543
432
54321
4321
543
432
54321
54321
;;0;0;3
202
1032
12
722
202
1032
12
722
2
min2221
06 ≥x .
07 ≥x .
08 ≥x
Thay 0; 444 ≥−= ttx
Thay 00; 22222 ≥′′≥′′′−′= xxxxx
Thay 00; 33333 ≥′′≥′′′−′= xxxxx
( ) ( ) ( ) ( )
( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) 0;;;;;;;0;0;3
202
1032
12
722
2
min00.02221
8763322451
433221
85433
743322
65433221
8765433221
≥′′′′′′≥≥
=−′′−′−′′−′+
=−+−′′−′
=++′′−′−′′−′−
=++−′′−′+′′−′−
→+++−−′′−′+′′−′−=
xxxxxxxtxx
dtxxxxx
cxxtxx
bxtxxxx
axxtxxxxx
xxxxtxxxxxxf
- 8 -
9. ( )0
8
0
7
0
6
0
5
0
4
0
3
0
3
0
2
0
2
0
1 ,,,,,,,,, xxxxtxxxxx ′′′′′′ ( ) 0
4
0
4
0
3
0
3
0
3
0
2
0
2
0
2
0
5
0
4
0
3
0
2
0
1 ,,,,,, txxxxxxxwithxxxxx −=′′−′=′′−′=
mibi ,1,0 =≥ )
( ) ( ) ( )
( )
( ) ( )njx
bxaxa
bxaxa
bxaxa
xcxf
j
m
n
n
j
jj
n
,13
2
maxmin1
0
111111
2111111
11111
1
=
=++
=++
=++
→=
≥
=
∑
0≥+inx
( ) min→xf
( ) max→xf
( ) ( ) ( )
( )
( ) ( )
=
+=
=+++
+
=+++
=+++
→±=
≥
+
+
+
=
+
=
∑∑
100
010
001
,13
2
maxmin1
11
2121
1111
0
11
222121
111111
11
nmnm
nn
nn
j
mmnnmnm
nnn
nn
m
i
in
n
j
jj
xaxa
xaxa
xaxa
A
mnjx
bxxaxa
bxxaxa
bxxaxa
xMxcxf
n
( )mix in ,1=+
- 9 -
10. ( ) ( )
( )
( )
−−=
=≥
=+
=+−−
=++
→−++=
8030
6140
5051
4,1;03
2883
1864
2555
2
max21
42
432
421
4321
A
jx
xx
xxx
xxx
xxxxxf
j
( ) ( )
( )
( )
−−=
=≥
=++
=++−−
=++
→−−−++=
108030
016140
005051
6,1;03
2883
1864
2555
2
max21
642
5432
421
654321
A
jx
xxx
xxxx
xxx
MxMxxxxxxf
j
a) ( )nxxxx ,,, 21 = ( )0,...,0,,...,, 21 nxxxx =
b) ( )00
2
0
1
0
,,, nxxxx = ( )0...,0,,,, 00
2
0
1
0
nxxxx =
c) ( )0...,0,,,, 00
2
0
1
0
nxxxx = ( )00
2
0
1
0
,,, nxxxx =
d)
1
1
1 2 0 2
2 0 7 4
3 8 0 3
4 1 6 2
5 9 2 0
1
2)
- 10 -
11. 0.55m 0.8m 0.45m
I: 1.2m 1
2
3
1
2
0
0
0
0
1
0
2
0.2
0.1
0.3
II: 1.5m 1
2
3
4
1
1
0
0
1
0
1
0
0
2
1
3
0.15
0.05
0.25
0.15
III: 1.8m 1
2
3
4
1
0
0
0
1
1
2
0
1
2
0
4
0
0.1
0.2
0
1)
( ) ( )
( )
( ) ( )5,103
20524
17432
2
min1
5431
5321
=≥
=+++
=+++
→
jx
xxxx
xxxx
xf
j
2)
( ) ( )
( )
( ) yytuxxxxx
xxxx
xxx
xxxxx
xf
′≥≤
=+−+
=−+
=++−+
→
54231
4321
432
54321
,;0;0,3
322
162
322
2
max1
3)
( ) ( )
( )
( ) ( )6,103
642
20524
17432
2
min1
651
5431
5321
=≥
=++−
=+++
=+++
→
jx
xxx
xxxx
xxxx
xf
j
4)
- 11 -
12. ( ) ( )
( )
( ) ( )4,103
832
152
52
2
max1
321
4321
321
=≥
≥++
=+++
≤−−
→
jx
xxx
xxxx
xxx
xf
j
5)
( ) ( )
( )
( ) ( )4,103
832
152
722
2
min1
431
432
321
=≥
≥++
≤++
−=−−
→
jx
xxx
xxx
xxx
xf
j
a)
b)
c)
- 12 -
13. §
( ) min→xf
Phương nvmmr ccccccc 121 +
iλ
nvmmr xxxxxxx 121 +
m
r
c
c
c
c
2
1
m
r
x
x
x
x
2
1
m
r
b
b
b
b
2
1 ( )
( )
( )
( ) mnmvmm
rnrvm
nvm
nvm
aaa
aaa
aaa
aaa
1
13
2212
1111
1000
0100
0010
0001
+
+
+
+ 1λ
2λ
rλ
( )xf 0f nvmmr ∆∆∆∆∆∆∆ + 121
∑∑
==
−=∆=
m
i
jijij
m
i
ii cacbcf
11
0 &
a) jj ∀≤∆ 0 ( ) 0fxf =
b) ( )miama ijj ,100 =≤>∆∃
vj
j
v xthi∆=∆ max
outxthenIf
awith
a
b
ri
i
r
iv
iv
i
i
λλ
λ
min
0
=
>=
1. Thay rx vx
2. rx )/ rva .
3. ×− iva
- 13 -
14. 4. ×∆− v
( ) max→xf
( ) ( ) min→−= xfxg
( ) ( )
( )
( )
( ) ( )
( )
( )
−−−
=
=≥
=++
=+++
=−−−
→+−+++=
→
=≥
≤+
=+++
=−−−
→−+++=
110030
0
2
3
2
1
120
092061
6,1;03
363
30
2
3
2
1
2
32926
2
min.04521
tan
5,1;03
363
30
2
3
2
1
2
32926
2
min4521
652
5432
5421
654321
52
5432
5421
54321
A
jx
xxx
xxxx
xxxx
xxxxxxxf
formdards
jx
xx
xxxx
xxxx
xxxxxxf
j
j
- 14 -
15. 2 5 4 1 -5 0
1x 2x 3x 4x 5x 6x
2
4
0
1x
3x
6x
32
30
36
1
0
0
-6
2
3
0
1
0
-2
½
0
-9
3/2
1
0
0
1
( )xf 184 0 -9 0 -3 -7 0
⇒∀≤∆ jj ,0 ( ) ( )0,0,30,0,32,,,, 54321 =xxxxx
( ) ( )
( )
( )
( ) ( )
( )
( )
−
−−
−−
=
=≥
=−++
=−+−
=+−+−
→+−++++=
→
=≥
=−++
−=+−
=+−+−
→+−++++=
312004
200102
101011
6,1;03
2324
922
15
2
min77361
tan
6,1;03
2324
922
15
2
min77361
6541
631
6421
654321
6541
631
6421
654321
A
jx
xxxx
xxx
xxxx
xxxxxxxf
formdards
jx
xxxx
xxx
xxxx
xxxxxxxf
j
j
- 15 -
16. 6 1 1 3 1 -7
1x 2x 3x 4x 5x 6x
1
1
1
2x
3x
5x
15
9
2
-1
-2
4
1
0
0
0
1
0
-1
0
2
0
0
1
(1)
-2
-3
( )xf 26+7 1∆
-5
2∆
0
3∆
0
4∆
-2
5∆
0
6∆
(3)
-7
1
1
6x
3x
5x
15
39
47
-1
-4
1
1
2
3
0
1
0
-1
-2
-1
0
0
1
1
0
0
( )xf -19+7 -2 -3 0 (1) 0 0
( )
( )outxa
inx
rv
j
2
66
1
3max
⇒=
⇒=∆=∆
{ } ⇒−−−<=∆=∆ 1,2,1,01max 44 ij abut
- 16 -
17. ( ) ( )
( )
( )
( ) ( )
( )
( )
=
=≥
=+
=++
=++
→−+−−=−=
→
=≥
=+
=++
=++
→+−++−=
10030
01240
00421
5,1;03
363
6024
5242
2
min32462)(1
tan
5,1;03
363
6024
522
2
max324621
52
432
321
54321
52
432
321
54321
A
jx
xx
xxx
xxx
xxxxxxfxg
formdards
jx
xx
xxx
xxx
xxxxxxf
j
j
2 -6 -4 2 -3
1x 2x 3x 4x 5x
2
2
-3
1x
4x
5x
52
60
36
1
0
0
2
4
3
(4)
2
0
0
1
0
0
0
1
( )xg 116 1∆
0
2∆
9
3∆
(16)
4∆
0
5∆
0
3x
4x
5x
13
34
36
1/4
-1/2
0
1/2
(3)
3
1
0
0
0
1
0
0
0
1
( )xg -92 -4 (1) 0 0 0
3x
2x
5x
22/3
34/3
2
1/3
-1/6
1/2
0
1
0
1
0
0
-1/6
1/3
-1
0
0
1
( )xg -310/3 -23/6 0 0 -1/3 0
- 17 -
19. §
1.
2.
3.
( ) ( )
( )
( )
( ) ( )
( )
( )
−
−−−
−−
=
=≥
=+++−
=+−−−
=+−−
→++++=
→
=≥
−
−−−
−−
=
=+++−
=−−−
=−−
→++=
00
10
01
5
1
3
4
3
2
3
1
1
25710
09300
7,1;03
3
2
3
1
3
4
3
2
3
1
5257
093
2
min5_21
tan
5,1;03
5
1
3
4
3
2
3
1
1
25710
09300
,
3
2
5
1
3
4
3
2
3
1
5257
093
2
min5_21
54321
75432
643
765421
54321
5432
43
5421
A
jx
xxxxx
xxxxx
xxx
MxMxxxxxxf
formdards
jx
A
xxxxx
xxxx
xx
xxxxxf
j
j
- 19 -
20. 1 2 0 1 -5
1x 2x 3x 4x 5x
M
M
1
6x
7x
1x
0
5
2/3
0
0
1
0
(1)
-1/3
-3
-7
2/3
-9
-5
4/3
0
-2
1/3
( )xf 5M+2/3
1∆
0
2∆
(-7/3+M)
3∆
2/3-10M
4∆
1/3-14M
5∆
16/3-2M
6x
2x
1x
0
5
7/3
0
0
1
0
1
0
-3
-7
-5/3
-9
-5
-1/3
0
-2
-1/3
( )xf 37/3 0 0 -47/3-3M -34/3-9M 2/3
( )
( )outxa
inxM
rv
i
j
7
2
22
1
1
5
min
3
7
max
⇒=
==
⇒+−=∆=∆
λλ
( ) ⇒≤
−−=>=∆=∆ 0
3
1
,2,00
3
2
max 55 ij abut
- 20 -
21. ( ) ( )
( )
( )
( ) ( )
( )
( )
−
−−
=
=≥
=++−
=+−−
→+++−=
→
=≥
−
−−
⇒
=+−
=+−−
→++−=
1
0
055
1
3
1
3
2
4,1;03
755
3
1
3
1
3
2
2
min97161
tan
3,1;03
055
1
3
1
3
2
755
3
1
3
1
3
2
2
min97161
421
321
4321
21
321
321
A
jx
xxx
xxx
Mxxxxxf
formdards
jx
xx
xxx
xxxxf
j
j
-16 7 9
1x 2x 3x
9
M
3x
4x
1/3
7
-2/3
-5
-1/3
(5)
1
0
( )xf 7M+3
1∆
10-5M
2∆
(-10+5M)
3∆
0
3x
2x
12/15
7/5
-1
-1
0
1
1
0
( )xf 17 0 0 0
( )
( )outxa
inxM
rv
i
j
4
2
22
5
5
7
min
510max
⇒=
==
⇒+−=∆=∆
λλ
⇒∀≤∆ jj ,0
( )
=
15
12
,
5
7
,0,, 321 xxx
170 =f
- 21 -
22. ( ) ( )
( )
( )
( ) ( )
( )
( )
( ) ( )
( )
( )
−−
−
=
=≥
=+−−
=+++
=++−
→+++−+=
→
−−
−
=
=≥
=+−−
=++
=+−
→+−+=
→
=≥
≤−−
=++
=+−
→−+=
00
10
01
1111
0212
0121
6,1;03
18
5022
272
2
min.02421
tan
1111
0212
0121
4,1;03
18
5022
272
2
min.02421
tan
3,1;03
18
5022
272
2
min2421
4321
6321
5321
654321
4321
321
321
4321
321
321
321
321
A
jx
xxxx
xxxx
xxxx
MxMxxxxxxf
formdards
A
jx
xxxx
xxx
xxx
xxxxxf
formdards
jx
xxx
xxx
xxx
xxxxf
j
j
j
- 22 -
23. 1 2 0 1
1x 2x 3x 4x
M
M
0
5x
6x
4x
27
50
18
1
2
1
-2
1
-1
1
(2)
-1
0
0
1
( )xf 77M
1∆
-2+3M
2∆
-4-M
3∆
(2+3M)
4∆
0
5x
3x
4x
2
25
43
0
1
2
-5/2
1/2
-1/2
0
1
0
0
0
1
( )xf -50+2M -4 -5-5M/2 0 0
( )
( )outxa
inxM
rv
i
j
6
2
33
2
25min
32max
⇒=
==
⇒+=∆=∆
λλ
⇒>=≤∆∀ 020 5xbutj
- 23 -
24. ( ) ( )
( )
( )
formdards
jx
A
xx
xx
xxxxf
j
tan
3,1;03
110
401
10
74
2
min221
32
31
321
→
=≥
=⇒
=+
=+
→+−=
2 -1 2
1x 2x 3x
2
-1
1x
2x
7
10
1
0
0
1
(4)
1
( )xf 4
1∆
0
2∆
0
3∆
(5)
3x
2x
7/4
33/4
1/4
-1/4
0
1
1
0
( )xf -19/4 -5/4 0 0
⇒∀≤∆ jj ,0
( )
=
4
7
,
4
33
,0,, 321 xxx
4
19
0 −=f
- 24 -
25. ( ) ( )
( )
( )
( ) ( ) ( ) min241
tan
3,1;03
011
110
5
8
2
max241
321
21
32
321
→+−−=−=
→
=≥
−
=⇒
=−
=+
→−+=
xxxxfxg
formdards
jx
A
xx
xx
xxxxf
j
-4 -1 2
1x 2x 3x
2
-4
3x
1x
8
5
0
1
(1)
-1
1
0
( )xg -4
1∆
0
2∆
(7)
3∆
0
2x
1x
8
13
0
1
1
0
1
1
( )xg -60 0 0 -7
⇒∀≤∆ jj ,0
( ) ( )0,8,13,, 321 =xxx
6000 =−= gf
- 25 -
26. ( ) ( )
( )
( )
( ) ( )
( )
( )
−
=
=≥
=+−+
=+++
→+++−=
→
=≥
≤−+
≤++
→+−=
10112
01121
5,1;03
102
122
2
min0021
tan
3,1;03
102
122
2
min21
5321
4321
54321
321
321
321
A
jx
xxxx
xxxx
xxxxxxf
formdards
jx
xxx
xxx
xxxxf
j
j
1 -2 1 0 0
1x 2x 3x 4x 5x
0
0
4x
5x
12
10
1
2
2
1
1
-1
1
0
0
1
( )xf 0
1∆
-1
2∆
2
3∆
-1
4∆
0
5∆
0
2x
5x
6
4
1/2
3/2
1
0
1/2
-3/2
1/2
-1/2
0
1
( )xf -12 -2 0 -2 -1 0
⇒∀≤∆ jj ,0
( ) ( )0,6,0,, 321 =xxx
120 −=f
- 26 -
27. ( ) ( )
( )
( )
( ) ( ) ( )
( )
( )
( ) ( ) ( )
( )
( )
−
−
==≥
=+−++
=+++−
→++++−−=−=
→
−
−
==≥
=−++
=+++−
→+++−−=−=
→
=≥
≥++
≤++−
→−+=
1
0
10131
01321
6,1;03
53
1032
2
min0021
tan
10131
01321
,5,1;03
53
1032
2
min0021
tan
3,1;03
53
1032
2
max21
65321
4321
654321
5321
4321
54321
321
321
321
Ajx
xxxxx
xxxx
Mxxxxxxxfxg
formdards
Ajx
xxxx
xxxx
xxxxxxfxg
formdards
jx
xxx
xxx
xxxxf
j
j
j
- 27 -
28. -1 -2 1 0 0
1x 2x 3x 4x 5x
0
M
4x
6x
10
5
-1
1
2
(3)
3
1
1
0
0
-1
( )xg 5M
1∆
M+1
2∆
(3M+2)
3∆
M-1
4∆
0
5∆
-M
4x
2x
20/3
5/3
-5/3
1/3
0
1
7/3
1/3
1
0
(2/3)
-1/3
( )xg -10/3 1/3 0 -5/3 0 (2/3)
5x
2x
10
5
-5/2
-1/2
0
1
7/2
3/2
3/2
1/2
1
0
( )xg -10 2 0 -4 -1 0
( ) ⇒≤
−−=>=∆∃ 0
2
1
;
2
5
,02 11 iabut
( ) ( )
( )
( )
( ) ( ) ( )
( )
( )
−
−
−
=⇒=≥
=++−
=++−
=−+
→+++−−=−=
→
=≥
−
−
−
=⇒
=+−
=+−
=−+
→−+=
10
01
00
230
5100
121
5,1;03
423
5510
22
2
min331
tan
3,1;03
230
5100
121
423
5510
22
2
max331
532
432
321
54321
32
32
321
321
Ajx
xxx
xxx
xxx
MxMxxxxxfxg
formdards
jx
A
xx
xx
xxx
xxxxf
j
j
-3 -1 3
- 28 -
30. ( ) ( )
( )
( )
( ) ( )
( )
( )
−−
−−
−−−
=⇒=≥
=+−−
=+−+−
=+−+−−
→++−+=
→
=≥
−−
−−
−−−
=⇒
=−−
=+−+−
=−+−−
→−+=
10
00
01
00
2
1
21
10122
01214
7,1;03
23
2
1
2
1022
1224
2
min21
tan
5,1;03
00
2
1
21
10122
01214
23
2
1
2
1022
1224
2
min21
7321
5321
64321
76321
321
5321
4321
321
Ajx
xxxx
xxxx
xxxxx
MxMxxxxxf
formdards
jx
A
xxx
xxxx
xxxx
xxxxf
j
j
2 1 -1 0 0
1x 2x 3x 4x 5x
M
0
M
6x
5x
7x
12
10
23
-4
-2
1
-1
2
-2
(2)
-1
-1/2
-1
0
0
1
0
0
( )xf 35M
1∆
-3M-2
2∆
-3M-1
3∆
(3M/2
+1)
4∆
-M
5∆
0
3x
5x
7x
6
16
26
-2
-4
0
-1/2
3/2
-9/4
1
0
0
-1/2
-1/2
-1/4
0
1
0
( )xf 26M-6 0 (-9M/4-1/2) 0 -M/4+1/2 0
⇒>=∀≤∆ 026,0 7xbutjj
- 30 -
31. ( ) ( )
( )
( )
( ) ( ) ( )
( )
( ) 5,1;03
2
23
732
2
min201
tan
5,1;03
11001
31100
32010
2
23
732
2
max201
541
543
542
54
541
543
542
54
=≥
=++
=−−
=++
→−−−=−=
→
=≥
−−=⇒
=++
=−−
=++
→++=
jx
xxx
xxx
xxx
xxxfxg
formdards
jx
A
xxx
xxx
xxx
xxxf
j
j
0 0 0 -1 -1
1x 2x 3x 4x 5x
0
0
0
2x
3x
1x
7
2
2
0
0
1
1
0
0
0
1
0
2
-1
(1)
3
-3
1
( )xg -20
1∆
0
2∆
0
3∆
0
4∆
(1)
5∆
1
2x
3x
4x
3
4
2
-2
1
1
1
0
0
0
1
0
0
0
1
1
-2
1
( )xg -22 -1 0 0 0 0
⇒∀≤∆ jj ,0
( ) ( )0,2,4,3,0,,,, 54321 =xxxxx 2200 =−= gf
- 31 -
32. ( ) ( )
( )
( )
( ) ( ) ( )
( )
( ) ( ) ( )
( )
( ) 7,1;03
10
01
00
00212
10211
01244
422
62
6244
2
min0021
tan
00212
10211
01244
422
62
6244
2
min0021
tan
3,1;03
422
62
6244
422
62
6244
2
max21
7321
65321
4321
7654321
321
5321
4321
54321
321
321
321
321
321
321
321
=≥
−
−
−−−
=⇒
=++−
=+−++
=+−−−
→+++++−−=−=
→
−
−
−−−
=⇒
=+−
=−++
=+−−−
→+++−−=−=
→
=≥
=+−
≥++
≤−−−
⇒
=+−
≥++
−≥++
→−+=
jx
A
xxxx
xxxxx
xxxx
MxMxxxxxxxfxg
formdards
A
xxx
xxxx
xxxx
xxxxxxfxg
formdards
jx
xxx
xxx
xxx
xxx
xxx
xxx
xxxxf
j
j
-1 -2 1 0 0
1x 2x 3x 4x 5x
0
M
M
4x
6x
7x
6
6
4
-4
1
2
-4
1
-1
-2
2
(2)
1
0
0
0
-1
0
( )xg 10M
1∆
3M+1
2∆
-2
3∆
(4M-1)
4∆
0
5∆
-M
4x
6x
3x
10
2
2
-2
-1
1
-5
(2)
-1/2
0
0
1
1
0
0
0
-1
0
( )xg 2M+2 -M+2 (2M+3/2) 0 0 -M
4x
2x
3x
15
1
5/2
-9/2
-1/2
3/4
0
1
0
0
0
1
1
0
0
-5/2
-1/2
-1/4
( )xg 1/2 11/4 0 0 0 3/4
⇒≤>=∆ 0,0
4
3
55 iabut
- 32 -
33. ( ) ( )
( )
( )
( ) ( ) ( )
( )
−
=⇒
=+++
=+++
=++−
→++−−−−=−=
→
=≥
−
⇒
=+++
=++
=+−
⇒
=+++
=++
−=−+−
→+++=
00
10
01
1121
0512
0321
102
2052
1532
2
min321
tan
4,1;03
1121
0512
0321
102
2052
1532
102
2052
1532
2
max321
4321
6321
5321
654321
4321
321
321
4321
321
321
4321
A
xxxx
xxxx
xxxx
MxMxxxxxxfxg
formdards
jx
xxxx
xxx
xxx
xxxx
xxx
xxx
xxxxxf
j
-1 -2 -3 -1
1x 2x 3x 4x
M
M
-1
5x
6x
4x
15
20
10
1
2
1
-2
1
2
3
(5)
1
0
0
1
( )xg 35M-10
1∆
3M
2∆
-M
3∆
(8M+2)
4∆
0
5x
3x
4x
3
4
6
-1/5
2/5
3/5
-13/5
1/5
9/5
0
1
0
0
0
1
( )xg 3M-18 -M/5-4/5 -13M/5-2/5 0 0
⇒>=∀≤∆ 03,0 5xbutjj
- 33 -
34. ( ) ( )
( )
( )
( ) ( )
( )
( ) 6,1;03
100112
001111
010111
182
27
15
2
min0021
tan
4,1;03
182
27
15
2
min21
6321
4321
5321
65421
321
4321
321
421
=≥
−−
−
⇒
=+−−
=+++
=+−+
→++++−=
→
=≥
≤−−
=+++
≤−+
→++−=
jx
xxxx
xxxx
xxxx
xxxxxxf
formdards
jx
xxx
xxxx
xxx
xxxxf
j
j
-2 1 0 1 0 0
1x 2x 3x 4x 5x 6x
0
1
0
5x
4x
6x
15
27
18
1
1
(2)
1
1
-1
-1
1
-1
0
1
0
1
0
0
0
0
1
( )xf 27
1∆
(3)
2∆
0
3∆
1
4∆
0
5∆
0
6∆
0
5x
4x
1x
6
18
9
0
0
1
3/2
3/2
-1/2
-1/2
(3/2)
-1/2
0
1
0
1
0
0
-1/2
-1/2
1/2
( )xf 0 0 3/2 (5/2) 0 0 -3/2
5x
3x
1x
12
12
15
0
0
1
2
1
0
0
1
0
1/3
2/3
1/3
1
0
0
-2/3
-1/3
1/3
( )xf -30 0 -1 0 -5/3 0 -2/3
⇒∀≤∆ jj ,0
( ) ( )0,12,0,15,,, 4321 =xxxx 300 −=f
- 34 -
35. ( ) ( )
( )
( )
( ) ( ) ( )
( )
( ) 7,1;03
1002820
0102050
0011031
6282
325
13
2
min000421
tan
4,1;03
6282
325
13
2
max421
7432
642
5421
7654321
432
42
421
4321
=≥
−−⇒
=+++
=+−−
=+++
→+++−−−−=−=
→
=≥
≤++
≤−−
≤++
→+++=
jx
xxxx
xxx
xxxx
xxxxxxxxfxg
formdards
jx
xxx
xx
xxx
xxxxxf
j
j
-2 -4 -1 -1 0 0 0
1x 2x 3x 4x 5x 6x 7x
-2
0
0
1x
6x
7x
1
3
6
1
0
0
3
-5
2
0
0
(8)
1
-2
2
1
0
0
0
1
0
0
0
1
( )xg -2
1∆
0
2∆
-2
3∆
(1)
4∆
-1
5∆
-2
6∆
0
7∆
0
1x
6x
3x
1
3
3/4
1
0
0
3
-5
1/4
0
0
1
1
-2
1/4
1
0
0
0
1
0
0
0
1/8
( )xg -11/4 0 -9/4 0 -5/4 -2 0 -1/8
⇒∀≤∆ jj ,0
( )
= 0,
4
3
,0,1,,, 4321 xxxx
4
11
00 =−= gf
- 35 -
36. ( ) ( )
( )
( )
( ) ( )
( )
( ) 8,1;03
10
01
00
436604
121102
210111
1843664
322
22
2
min542223101
436604
121102
210111
tan
6,1;03
1843664
322
22
2
min542223101
865431
765431
65321
87654321
65431
65431
65321
654321
=≥
−
−
−
⇒
=+++−+
=+++−+
=++−+
→+++−−++−−=
−
−
−
⇒
→
=≥
=++−+
=++−+
=++−+
→+−−++−−=
jx
xxxxxx
xxxxxx
xxxxx
MxMxxxxxxxxf
formdards
jx
xxxxx
xxxxx
xxxxx
xxxxxxxf
j
j
-10 -3 2 2 -2 -4
1x 2x 3x 4x 5x 6x
-3
M
M
2x
7x
8x
2
3
18
1
2
4
1
0
0
-1
(1)
6
0
-1
-6
1
2
3
2
1
4
( )xg 21M-1
1∆
6M+7
2∆
0
3∆
(7M+1)
4∆
-7M-2
5∆
5M-1
6∆
5M-2
2x
3x
8x
5
3
0
3
2
-8
1
0
0
0
1
0
-1
-1
0
3
2
-9
3
1
-2
( )xg -4 -8M+5 0 0 -1 -9M-3 -2M-3
⇒∀≤∆ jj ,0
( ) ( )0,0,0,3,5,0,,,,, 654321 =xxxxxx 40 −=f
- 36 -
37. ( ) ( )
( )
( ) 6,1;03
1843664
322
22
2
max24223101
65431
65431
65321
654321
=≥
=++−+
=++−+
=++−+
→+++−−+=
jx
xxxxx
xxxxx
xxxxx
xxxxxxxf
j
( ) ( )
( )
( )
( ) ( )0,0,2,0,,,:
4,1;03
532
273
22
2
min21
4321
43
432
4321
4321
==
=≥
≤+
≤++−
=−−+
→+−+=
xxxxxĐS
jx
xx
xxx
xxxx
xxxxxf
j
( ) ( )
( )
( )
( )
==
=≥
≤+−
≥+−
=+++
→−−−=
3
1
,0,
3
2
,0,,,:
4,1;03
424
132
2423
2
min4321
4321
431
431
4321
4321
xxxxxĐS
jx
xxx
xxx
xxxx
xxxxxf
j
( ) ( )
( )
( )
( ) ( )4,0,2,,:
3,1;03
1423
52
62
2
min231
321
321
321
321
321
==
=≥
=+−
≤−+
=++−
→++=
xxxxĐS
jx
xxx
xxx
xxx
xxxxf
j
- 37 -
38. ( ) ( )
( )
( ) 4,1;03
1242
1623
57326
2
min2341
321
321
4321
4321
=≥
=−+
≤++
=++−
→−−−=
jx
xxx
xxx
xxxx
xxxxxf
j
( ) ( )
( )
( )
( ) ( )0,1,2,0,,,:
4,1;03
12
532
8423
2
min221
4321
321
321
4321
4321
==
=≥
≤−+
−≥+−
=+++
→+−−=
xxxxxĐS
jx
xxx
xxx
xxxx
xxxxxf
j
( ) ( )
( )
( )
PATUnoĐS
jx
xxx
xxxx
xxx
xxxxxf
j
:
4,1;03
03
422
1623
2
max641021
432
4321
421
4321
=≥
≥−+
−≥−++
≤+−
→−+−=
( ) ( )
( )
( )
( )
==
=≥
=+−
≥++
≤−+−
→+−−=
5
2
,
5
12
,
5
14
,,:
3,1;03
422
62
624
2
min21
321
321
321
321
321
xxxxĐS
jx
xxx
xxx
xxx
xxxxf
j
- 38 -
39. §
(P)
( ) ( )
( )
( ) ( )3,13
2
min1
0
2323222121
1313212111
332211
=
⇒
=++
=++
→++=
≥ jx
bxaxaxa
bxaxaxa
xcxcxcxf
j
=⇒
=
2313
2212
2111
232221
131211
aa
aa
aa
A
aaa
aaa
A
T
(D)
( ) ( )
( )
( ) ( ) yytujy
cyaya
cyaya
cyaya
ybybyg
j ′=
≤+
≤+
≤+
→+=
2,13
3
2
max1
32113
2222112
1221111
2211
1. ( ) min→xPf ( ) max→yDg
2.
3. ibic &
4.
( ) min
1
→= ∑
=
n
j
jj xcxf ( ) max
1
→= ∑
=
m
i
ii ybyg
∑
=
n
j
jij xa
1
=
≥
≤
i
i
i
b
b
b
′
≥
≤
yytu
yi
0
0
′
≤
≥
yytu
x j
0
0
∑
=
m
i
iij ya
1
=
≥
≤
j
j
j
c
c
c
- 39 -
40. ( ) ( )
( )
( )
( )
( )
( )
( )
( ) ( )
( )
( ) 0,,03
1
132
3
2512
2
min20751
111
321
111
512
1315
1211
1112
,0,0,3
32035
272
152
2
max321
321
321
321
321
321
321
4321
4321
4321
4321
4321
≤′≥
=++
−≤++−
≥++−
≥++
→++=
−
−
=⇒
−−
=
′≤≥
≥+++
=+++
≤+−−
→+−+=
yyytuyy
yyy
yyy
yyy
yyy
yyyyg
D
AA
yytuxxxx
xxxx
xxxx
xxxx
xxxxxf
T
- 40 -
41. ( ) ( )
( )
( )
( )
( ) ( )
( )
( ) 0,0,3
64
33
322
12
533
2
max63010801
0114
0113
1212
0211
1313
00101
11223
11111
43213
,,0,,03
6
30223
10
804323
2
min63251
4321
321
321
4321
321
4321
4321
54321
31
54321
54321
54321
54321
≤≥′
−=++
=−+
≤+−+
−≤−+−
≥+++
→++−=
−
−
−−
=⇒
−−−
−
=
′≥≤
≤+
≤+−−−
−≥++++
=+++−
→−++−=
yyyyytuy
yyy
yyy
yyyy
yyy
yyyy
yyyyyg
D
AA
yytuxxxxx
xx
xxxxx
xxxxx
xxxxx
xxxxxxf
T
- 41 -
42. §
1.
2.
3. ( ) ( )DofyPofx 00
& ( ) ( )00
ygxf =
( ) ( )DofyPofx 00
&
( )
( )
==
−
==
−
∑
∑
=
=
mibxay
njcyax
n
j
jjiji
m
i
jiijj
,10
,10
1
00
1
00
( )00
2
0
1
0
,,, myyyy =
∑ =⇒>
j
ijiji bxay 00
( )00
2
0
1
0
,,, myyyy = ∑
=
−
m
i
jiij cya
1
0
( )nj ,1=
0=jx
( ) ( )
( )
( ) ( )5,103
363
30
2
3
2
1
2
32926
2
min54521
52
532
521
5321
4
4
4
=≥
≤+
=++
=−−
→+++=
+
−
−
jx
xx
xxxx
xxxx
xxxxxxf
j
- 42 -
43. ( ) ( )
( )
( ) 0,,3
5
2
3
9
1
2
1
2
4
5326
2
2
max3630321
321
321
21
321
1
321
2
≤′
−≤++−
≤+−
≤
≤++−
≤
→++=
yyytuyy
yyy
yy
y
yyy
y
yyyyg
( ) ( ) 1840,0,30,0,32 00
== xfwithx
4230
232
0
3
1
0
1
=>⇒=
=>⇒=
yx
yx
( )0,0,30,0,320
=x 00363 352 =⇒<−+ yxx
( ) ( ) 184&0,4,2 00
== ygy
( ) ( )
( )
( ) ( )3,13
3
2
424
6342
2
2
min3660521
3
2
21
321
1
321
=′
≥
−≥
≥+
≥++
−≥
→++=
jyytux
x
x
xx
xxx
x
xxxxf
j
=⇒
=⇒
10030
01240
00421
100
010
024
342
001
T
AA
- 43 -
44. ( ) ( )
( )
( ) ( )5,103
363
6024
5242
2
max324621
52
432
321
54321
=≥
=+
=++
=++
→+−++−=
iy
yy
yyy
yyy
yyyyyyg
i
( ) 3
310
,2,0,
3
22
,
3
34
,0 00
=
= ygy
302
3
3
5
6
11
3
424
6342
4240
3
22
63420
3
34
3
0
5
3
2
1
3
21
321
21
0
3
321
0
2
=⇒>=
=
−=
=
⇔
=
=+
=++
⇒=+⇒>=
=++⇒>=
xy
x
x
x
x
xx
xxx
xxy
xxxy
( )
−== 3,
3
5
,
6
11
,, 321
0
xxxx
1)
( ) ( )
( )
( ) ( )4,103
34
325
13
2
max5421
432
2
21
5321
4
4
4
=≥
≤++
≤−−
≤++
→+++= −
jx
xxx
xx
xxx
xxxxxxf
j
a)
b)
- 44 -
45. 2)
( ) ( )
( )
( ) 0,,3
22
42
22
2
max1850271
321
321
321
21
321
3
≤′
−≤−+
≤−+−
≤++
→++=
xyytuxx
xxx
xxx
xxx
xxxxf
c)
d)
3)
( ) ( )
( )
( ) ( )4,103
182
27
15
2
min21
321
21
21
21
43
3
4
=≥
≤−−
=+++
≤−+
→++−=
jx
xxx
xxxx
xxx
xxxxf
j
e)
f)
4)
( ) ( )
( )
( ) ( )3,103
2022
152
12
2
max3321
321
21
21
321
3
3
=≥
≤++
≤++
≤++
→++=
jx
xxx
xxx
xxx
xxxxf
j
g)
h)
- 45 -
46. 1)
( ) ( )
( )
( )
( )
( )
( ) ( )4,103
36282
2325
113
2
max421
432
2
21
321
4
4
4
=≥
≤++
≤−−
≤++
→+++=
jx
xxx
xx
xxx
xxxxxf
j
i)
j)
−
−
=⇒
−−=
221
800
253
001
2820
2050
1031
T
AA
( ) ( )
( )
( ) ( )3,1,03
122
18
4253
2
2
min631
321
321
1
321
3
=≥
≥+−
≥
≥+−
≥
→++=
jy
yyy
y
yyy
y
yyyyg
j
( ) ( ) 4
11
0,
4
3
,0,1,,, 0
4321
0
=
== xfwithxxxxx
8
1
4
3
21
3
0
3
1
0
1
=>⇒=
=>⇒=
yx
yx
= 0,
4
3
,0,10
x 003325 242 =⇒≠−=−−− yxx
( ) 4
11
&
3
1
,0,2 00
=
= ygy
2)
- 46 -
47. ( ) ( )
( )
( ) 6,1;03
1843664
322
22
2
min542223101
65431
65431
65321
654321
=≥
=++−+
=++−+
=++−+
→+−−++−−=
jx
xxxxx
xxxxx
xxxxx
xxxxxxxf
j
k)
l)
−−
−
=⇒
−
−
−
=
412
321
610
611
001
421
436604
121102
210111
T
AA
( ) ( )
( )
( ) ( )3,13
442
232
26
26
3
1042
2
max18321
321
321
32
321
1
321
321
=′
−≤++
−≤++
≤−−
≤++−
−≤
−≤++
→++=
jyytuy
yyy
yyy
yy
yyy
y
yyy
yyyyg
j
( ) ( ) ( ) 4
11
0,0,0,3,5,0,,,,, 0
654321
0
=== xfwithxxxxxxx
−−=
=
−=
⇒
−=+⇒>=
−=⇒>=
16
3
1603
305
2
3
1
32
0
3
1
0
2
ay
ay
y
yyx
yx
( ) ( ) ( ) 4&,16,,3,, 0
321
0
−=∈∀−−−== ygRaaayyyy
3)
- 47 -
48. ( ) ( )
( )
( )
( )
( )
( ) ( )3,103
3
4
1
243
122
2
min331
3
21
21
321
3
=≥
≥
≥++
≥+
→++=
jx
x
xxx
xx
xxxxf
j
m)
n)
=⇒
=
1110
0012
1031
101
100
113
021
T
AA
( ) ( )
( )
( ) ( )3,1,03
3
32
13
2
max
4
1
421
32
21
21
4321
=≥
≤+
≤+
≤+
→+++=
jy
yy
yy
yy
yyyyyg
j
( ) ( ) ( ) 4
11
0,3,0,1,,, 0
4321
0
=== ygwithyyyyy
4
1
03
2201
3
0
3
21
0
1
=⇒>=
=+⇒>=
xy
xxy
( )0,3,0,10
=y 00132 221 =⇒≠−=−+ xyy
( ) 4
11
&
4
1
,0,2 00
=
= xfx
4)
( ) ( )
( )
( )
( )
( )
( ) ( )4,103
3164322
230322
115232
2
max2731
4321
321
321
321
4
4
=≥
=+−−
≤+−
≤+−−
→−+−=
jx
xxxx
xxx
xxxx
xxxxxf
j
o)
- 48 -
49. p)
−−
−−−
=⇒
−−
−
−−
=
402
331
223
111
4321
0321
2131
T
AA
( ) ( )
( )
( ) yytuyyy
yy
yyy
yyy
yyy
yyyyg
′≥
−≥+
≥−+−
−≥−−−
≥++
→++=
321
31
321
321
321
321
,0,3
242
133
7223
3222
2
min1630151
( ) ( ) 20
2
1
,0,0,7,,, 0
4321
0
=
== xfwithxxxxx
2420
2
1
322207
31
0
4
321
0
1
−=+⇒>=
=++⇒>=
yyx
yyyx
=
2
1
,0,0,70
x 001630322 2321 =⇒≠−=−+− yxxx
( ) 20&
2
5
,0,4 00
=
−= ygy
5)
( ) ( )
( )
( )
( )
( )
( ) ( )4,103
31222
210322
11523
2
min879121
4321
4321
321
321
4
4
=≥
≥+++
=+++
≤+++
→+++=
jx
xxxx
xxxx
xxxx
xxxxxf
j
q)
r)
- 49 -
50.
=⇒
=
131
221
122
113
1211
3221
1123
T
AA
( ) ( )
( )
( ) 0,,03
83
722
922
1223
2
max1210151
321
321
321
321
321
321
≥′≤
≤++
≤++
≤++
≤++
→++=
yyytuyy
yyy
yyy
yyy
yyy
yyyyg
( ) ( ) ( ) 520,4,0,2,,, 0
4321
0
=== xfwithxxxxx
72204
122302
321
0
3
321
0
1
=++⇒>=
=++⇒>=
yyyx
yyyx
( )0,4,0,20
=x 0051523 14321 =⇒≠−=−+++ yxxxx
( ) 52&
2
17
,5,0 00
=
−= ygy
- 50 -
51. 6)
( ) ( )
( )
( )
( )
( )
( ) ( )3,103
320
2602
1402
2
max3341
1
321
321
321
=≥
≤
≤++
≤++
→++=
jx
x
xxx
xxx
xxxxf
j
s)
t)
=⇒
=
021
012
111
001
211
121
T
AA
( ) ( )
( )
( ) 3,1,03
32
32
4
2
min2060401
21
21
321
321
=≥
≥+
≥+
≥++
→++=
jy
yy
yy
yyy
yyyyg
j
( ) ( ) ( ) 14020,0,20,, 0
321
0
=== xfwithxxxx
32020
4020
21
0
3
321
0
1
=+⇒>=
=++⇒>=
yyx
yyyx
( )20,0,200
=x
Raa
ay
ay
ay
jy j ∈>
=
+−=
−=
⇒=≥ ,0,1
25
3,1,0
3
2
1
( ) ( ) 140&,1,25 00
=+−−= ygaaay
- 51 -
52. 7)
( ) ( )
( )
( )
( )
( )
( ) ( )3,103
330526
2923
120353
2
min861
3
3
21
21
321
321
=≥
≥++
≥++
≥++
→++=
jx
xxx
xxx
xxx
xxxxf
j
u)
v)
=⇒
=
523
235
613
526
231
353
T
AA
( ) ( )
( )
( ) ( )3,1,03
1523
8235
663
2
max309201
321
321
321
321
=≥
≤++
≤++
≤++
→++=
jy
yyy
yyy
yyy
yyyyg
j
( ) ( ) 3
20
0,0,
3
1
,, 0
321
0
=
== ygwithyyyy
203530
3
1
321
0
1 =++⇒>= xxxy
= 0,0,
3
10
y
005663
00
3
19
8235
1321
2321
=⇒≠−=−++
=⇒≠
−
=−++
xyyy
xyyy
( ) 3
20
&
3
20
,0,0 00
=
= xfx
8)
- 52 -
53. ( ) ( )
( )
( )
( )
( )
( ) ( )3,103
38
21023
11243
2
min21
3
3
21
21
321
321
=≥
−≥−−
≥++
≥+−
→++=
jx
xxx
xxx
xxx
xxxxf
j
w)
x)
−
−−=⇒
−−
−
=
124
113
131
111
213
431
T
AA
( ) ( )
( )
( ) ( )3,1,03
124
23
13
2
max810121
321
321
321
321
=≥
≤−+
≤−+−
≤++
→−+=
jy
yyy
yyy
yyy
yyyyg
j
( ) ( ) 5
21
0,
10
3
,
10
1
,, 0
321
0
=
== ygwithyyyy
10230
10
3
12430
10
1
321
0
2
321
0
1
=++⇒>=
=+−⇒>=
xxxy
xxxy
= 0,
10
3
,
10
10
y
=+
=+
⇒=⇒≠−=−−+−
1023
124
00223
31
31
2321
xx
xx
xyyy
( ) 5
21
&
5
13
,0,
5
8 00
=
= xfx
- 53 -
54. 9)
( ) ( )
( )
( )
( )
( )
( ) ( )3,103
3232
26322
143
2
min421
3
3
21
21
321
321
=≥
≥++−
≥−+
≥+−
→++=
jx
xxx
xxx
xxx
xxxxf
j
y)
z)
−
−
−
=⇒
−
−
−
=
333
221
121
321
322
311
T
AA
( ) ( )
( )
( ) ( )3,1,03
4333
222
12
2
max2641
321
321
321
321
=≥
≤+−
≤++−
≤−+
→++=
jy
yyy
yyy
yyy
yyyyg
j
( ) ( ) 33
349
33
17
,
33
31
,
11
10
,, 0
321
0
=
== ygwithyyyy
=
=
=
=
⇒
=++−⇒>=
=−+⇒>=
=+−⇒>=
33
26
,
11
14
,
11
32
33
26
11
14
11
32
2320
33
17
63220
33
31
430
11
10
0
3
2
1
321
0
2
321
0
2
321
0
1
xPATU
x
x
x
xxxy
xxxy
xxxy
10)
- 54 -
55. ( ) ( )
( )
( )
( )
( )
( ) ( )3,103
31235
21434
116223
2
min48161
3
3
21
21
321
321
=≥
≥++
≥++
≥++
→++=
jx
xxx
xxx
xxx
xxxxf
j
aa)
=⇒
=
112
332
543
135
134
223
T
AA
( ) ( )
( )
( ) ( )3,1,03
42
8332
16543
2
max1214161
321
321
321
321
=≥
≤++
≤++
≤++
→+=
jy
yyy
yyy
yyy
yyyyg
j
( ) ( ) ( ) 440,2,1,, 0
321
0
=== ygwithyyyy
143402
1622301
321
0
2
321
0
1
=++⇒>=
=++⇒>=
xxxy
xxxy
( )0,2,10
=y
=+
=+
⇒=⇒≠−=−+
1423
1622
00516543
32
32
1321
xx
xx
xyyy
( ) ( ) 44&5,3,0 00
== xfx
- 55 -
56. §
mAAA ,...,, 21 maaa ,...,, 21 nBBB ,...,, 21 nbbb ,...,, 21 .
( )miAi ,1= ( )njB j ,1=
∑∑ =
j
j
i
i ba
[ ] nmijcC
×
= ijc iA jB
ijx iA jB
( ) ( )
( )
( )
( )
( ) ( )njmix
njbx
miax
xcxf
ij
m
i
jij
n
j
iij
i j
ijij
,1;,103
,1
,1
2
min1
1
1
==≥
==
==
→=
∑
∑
∑∑
=
=
- 56 -
57. Thu 1
1
b
B
2
2
b
B …
j
j
b
B …
n
n
b
B
11 : aA 11
11
x
c
12
12
x
c …
j
j
x
c
1
1 …
n
n
x
c
1
1
22 : aA 21
21
x
c
22
22
x
c …
j
j
x
c
2
2 …
n
n
x
c
2
2
… … … … … … …
ii aA : 1
1
i
i
x
c
2
2
i
i
x
c
ij
ij
x
c
in
in
x
c
… … … … … … …
mm aA : 1
1
m
m
x
c
2
2
m
m
x
c
mj
mj
x
c
mn
mn
x
c
a.
b.
c. 0>ijx
d.
- 57 -
58. §
( )ji, ( )ji, 11 122
Thu
20
1B
40
2B
30
3B
30:1A
20
1
×
10
3
×
5
25:2A
5 4
25
2
×
35:3A
8
30
5
×
5
4
×
- 58 -
59. Thu
25
1B
25
2B
10
3B
10:1A
5 3
10
1
×
30:2A
25
7
×
5
6
× ×
8
20:3A
3
20
2
×
2
( ) nmijcC
×
= ( )miyyturi ,1=′ j ( )njyytus j ,1=′
( ) jiijijnmij srccwithcC ++=′′=′
×
( )miri ,1= j ( )njs j ,1= ji sr , 0=++=′ jiijij srcc
1) 0≥
2)
1) ( )∗∗
ji ,
2) ( )∗∗
ji ,
3) ( )∗∗
ji ,
C
V
L
V
4)
( )
00
,
min
jiij
Vji
xx
C
=
∈
=′X ( ) nmijx
×
′
( )
( )
( )
∉
∈+
∈−
=′
Vjiifx
Vjiifxx
Vjiifxx
x
ij
L
jiij
C
jiij
ij
,
,
,
00
00
- 59 -
60. • 00
ji
x .
• 00
ji
x .
•
Thu 80
1B
20
2B
60
3B
50:1A
5 4
50
1
×
1r =
6
40:2A
20
3
×
20
2
×
6
2r =0
70:3A
60
7
×
9
10
11
×
3r =-4
1s =-3 2s =-2 3s =-7
2
6
4
7
3
0 2
1
3
3
1
2 −=⇒
=
−=
⇒
−=
−=
⇒= s
r
r
s
s
r
jiijij srcc ++=′
Ô (1,1):10-7+5=8.
Ô (1,2): 6-2+4=8.
Ô (2,2): 6-7+1=0
Ô (3,2): -4+9-2=2.
Ô (2,3): 0+6-7=-1
8 8
50
0
×
1−=r
20
0
×
20
0
×
1−
0=r
60
0
×
2
10
0
×
0=r
S=0 S=0 S=1
- 60 -
61. ( ) ( ) ( ) ( ){ },3,3,1,3,1,2,3,2=V
{ } 1010,20min =
- 61 -
63. §
∑ ∑> ji ba
( ) ( )
( )
( )
( )
( ) ( )njmix
njbx
miax
xcxf
ij
m
i
jij
n
j
iij
i j
ijij
,1;,103
,1
,1
2
min1
1
1
==≥
==
=≤
→=
∑
∑
∑∑
=
=
∑ ∑< ji ba
( ) ( )
( )
( )
( )
( ) ( )njmix
njbx
miax
xcxf
ij
m
i
jij
n
j
iij
i j
ijij
,1;,103
,1
,1
2
min1
1
1
==≥
=≤
==
→=
∑
∑
∑∑
=
=
1+nB 01 >−= ∑ ∑+ jin bab
1+mA 01 >−= ∑ ∑+ ijm aba
- 63 -
64. Thu 20 40 60
80 3 4 1
30 4 2 3
50 1 5 6
Thu
20 40 60 (40)
Thu
80 3 4 4 0
10
1 0
60
0 0
10
R=0
30 4 7 2 0
30
3 4 0 2 R=2
50 1 0
20
5 1 6 5 0 0
30
R=0
S=1 S=-4 S=-1 S=0
0≥
=
300020
00300
1060100
X
( ) 180=xf
- 64 -
65. 80 20 60
50 5 4 1
40 3 2 6
70 7 9 11
80 20 60
50 5 8 4 8 1 0
50
R=6
40 3 0
20(-10)
2 0
20
6 -1
(+10)
R=0
70 7 0
60(+10)
9 11 0
10(-10)
R=-4
S=-3 S=-2 S=-7
( ) ( ) ( ) ( ){ },3,3,1,3,1,2,3,2=V
{ } 1010,20min =
- 65 -
66. 5 7 4 7 1 0
50
R=-1
3 0
10
2 0
20
6 0
10
R=0
7 0
70
9 3 11 1
0
R=0
S=0 S=0 S=1
0≥
=
0070
102010
5000
X
( ) 670=xf
60 70 40 30
100 2 1 4 3
80 5 3 2 6
20 6 2 1 5
60 70 40 30
100 2 0
30(+30)
1 0
70(-30)
4 5 3 0 R=3
80 5 0
30(-30)
3 -1
(+30)
2 0
20
6 0
30
R=0
20 6 2 2 -1 1 0 5 0 R=1
- 66 -
67. 20
S=-5 S=-4 S=-2 S=-6
( ) ( ) ( ) ( ){ },1,2,1,1,2,1,2,2=V
{ } 3070,30min =
2 0
60
1 0
40(-30)
4 4 3 -1
(+30)
R=-1
5 1
0
3 0
30(+30)
2 0
20
6 0
30 (-30)
R=0
6 3 2 0 1 0
20
5 0 R=0
S=1 S=1 S=0 S=0
( ) ( ) ( ) ( ){ },4,2,2,2,2,1,4,1=V
{ } 3040,30min =
2 0
60
1 0
10
4 4 3 0
30
R=0
5 1
0
3 0
60
2 0
20
6 1
0
R=0
6 3 2 0 1 0
20
5 1 R=0
S=0 S=0 S=0 S=1
0≥
=
02000
020600
3001060
X
( ) 460=xf
- 67 -
68. 20 100 145 30 150
120 6 3 1 4 5
150 1 2 5 4 3
150 2 4 3 1 6
25 3 1 4 2 7
20 100 145 30 150
120 6 4 3 0 1 0
120
4 5 5 1 R=2
150 1 0
20(-20)
2 0
75
5 5 4 6 3 0
55(+20)
R=3
150 2 -2
(+20)
4 -1 3 0
25
1 0
30
6 0
95(-20)
R=0
25 3 3 1 0
25
4 5 2 5 7 5 R=4
S=-4 S=-5 S=-3 S=-1 S=-6 S=-4
( ) ( ) ( ) ( ){ },1,2,5,2,5,3,1,3=V
{ } 2095,20min =
- 68 -
69. 6 6 3 0 1 0
120
4 5 5 1 R=0
1 2
0
2 0
75(-75)
5 5 4 6 3 0
75(+75)
R=0
2 0
20
4 -1
(+75)
3 0
25
1 0
30
6 0
75(-75)
R=0
3 5 1 0
25
4 5 2 5 7 5 R=0
S=2 S=0 S=0 S=0 S=0
( ) ( ) ( ) ( ){ },2,2,5,2,5,3,2,3=V
{ } 7575,75min =
6 6 3 1 1 0
120
4 5 5 1 R=0
1 2
0
2 1
0
5 5 4 6 3 0
150
R=0
2 0
20
4 0
75
3 0
25
1 0
30
6 0
0
R=0
3 4 1 0
25
4 4 2 4 7 4 R=-1
S=0 S=1 S=0 S=0 S=0
0≥
=
000250
030257520
1500000
0012000
X
( ) 1040=xf
- 69 -
70. 10 10 10 20 20
5 5 1 4 6 7
15 3 4 2 7 8
20 4 3 1 7 9
30 6 5 4 9 11
10 10 10 20 20
5 5 4 1 0
5
4 5 6 1 7 0 R=4
15 3 0
10
4 1 2 1 7 0
5(-5)
8 -1
(+5)
R=2
20 4 1 3 0 1 0
10
7 0
10
9 0 R=2
30 6 1 5 0
5
4 1 9 0
5(+5)
11 0
20(-5)
R=0
S=-5 S=-5 S=-3 S=-9 S=-11
( ) ( ) ( ) ( ){ },5,4,4,4,4,2,5,2=V
{ } 55,20min =
- 70 -
71. 5 3 1 0
5
4 5 6 1 7 0 R=0
3 0
10
4 2 2 2 7 0
0
8 0
5
R=1
4 0 3 0 1 0
10
7 0
10
9 0 R=0
6 0 5 0
5
4 1 9 0
10
11 0
15
R=0
S=-1 S=0 S=0 S=0 S=0
0≥
=
1510050
0101000
500010
00050
X
( ) 435=xf
30 15 2 15
25 3 4 2 6
15 5 1 6 2
40 2 1 5 3
- 71 -
72. 30 15 2 15
25 3 0
5
4 2 2 0
20
6 2 R=-1
15 5 3 1 0
15(-15)
6 5 2 -1
(+15)
R=0
40 2 0
25
1 0
0(+15)
5 4 3 0
15(-15)
R=0
S=-2 S=-1 S=-1 S=-3
( ) ( ) ( ) ( ){ },4,3,2,3,2,2,4,2=V
{ } 1515,15min =
3 0
5
4 2 2 0
20
6 3 R=0
5 3 1 0
0
6 5 2 0
15
R=0
2 0
25
1 0
15
5 4 3 0
0
R=0
S=0 S=0 S=0 S=1
0≥
=
001525
15000
02005
X
( ) 150=xf
- 72 -
73. 180 200 230 280
280 8 6 14 7
320 2 4 6 7
290 5 3 4 9
180 200 230 280
280 8 5 6 0
0
14 7 7 0
280
R=-3
320 2 0
180
4 -1 6 0
140
7 1 R=-2
290 5 5 3 0
200
4 0
90
9 5 R=0
S=0 S=-3 S=-4 S=-4
( ) ( ) ( ) ( ){ },3,2,3,3,2,3,2,2=V
{ } 140140,200min =
8 4 6 0
0
14 7 7 0
280
R=0
2 0
180
4 0
140
6 1
0
7 2 R=1
5 4 3 0
60
4 0
230
9 5 R=0
S=-1 S=0 S=0 S=0
- 73 -
74. 0≥
=
0230600
00140180
280000
X
( ) 3980=xf
8 7 12 15
10 8 9 12 5
19 4 8 5 9
11 5 9 7 1
9 1 2 6 3
8 7 12 15
10 8 1 9 1 12 7 5 0
4
0 0
6
R=0
19 4 -3
(+6)
8 0
6(-6)
5 0
12
9 4 0 0
1
R=0
11 5 2 9 5 7 6 1 0
11
0 4 R=4
9 1 0
8(-6)
2 0
1(+6)
6 7 3 4 0 6 R=6
S=-7 S=-8 S=-5 S=-5 S=0
( ) ( ) ( ) ( ){ },2,2,2,4,1,4,1,2=V
{ } 68,6min =
- 74 -
75. 8 4 9 4 12 7 5 0
4
0 0
6
R=0
4 0
6
8 3
0
5 0
12
9 4 0 0
1
R=0
5 5 9 8 7 6 1 0
11
0 4 R=0
1 0
2
2 0
7
6 4 3 1 0 3 R==-3
S=3 S=3 S=0 S=0 S=0
0≥
=
00872
011000
101206
64000
X
( ) 131=xf
20 50 60 30
50 4 5 1 0
40 2 3 6 0
70 9 7 11 0
- 75 -
76. 20 50 60 30
50 4 8 5 8 1 0
50
0 10 R=10
40 2 0
20
3 0
20(-10)
6 -1
(+10)
0 4 R=4
70 9 3 7 0
30(+10)
11 0
10(-10)
0 0
30
R=0
S=-6 S=-7 S=-11 S=0
( ) ( ) ( ) ( ){ },3,3,2,3,2,2,3,2=V
{ } 1020,10min =
4 7 5 7 1 0
50
0 9 R=-1
2 0
20
3 0
10
6 0
10
0 4 R=0
9 3 7 0
40
11 1
0
0 0
30
R=0
S=0 S=0 S=1 S=0
0≥
=
300400
0101020
05000
X
( ) 460=xf
- 76 -
77. 30 40 60 70
100 4 5 3 2
80 7 3 6 4
20 6 2 7 3
30 40 60 70
100 4 0 5 5 3 0
30(+30)
2 0
70(-30)
R=3
80 7 0
30
3 0
20
6 0
30(-30)
4 -1
(+30)
R=0
20 6 0 2 0
20
7 2 3 -1 R=1
S=-7 S=-3 S=-6 S=-5
( ) ( ) ( ) ( ){ },3,2,3,1,4,1,4,2=V
{ } 3030,70min =
4 -1
(+30)
5 4 3 0
60
2 0
40(-30)
R=-1
7 0
30(-30)
3 0
20
6 1
0
4 0
30(+30)
R=0
6 0 2 0
20
7 3 3 0 R=0
S=0 S=0 S=1 S=1
( ) ( ) ( ) ( ){ },4,1,4,2,1,2,1,1=V
- 77 -
79. 4 0
30
5 4 3 0
60
2 0
10
R=0
7 1
0
3 0
20
6 1
0
4 0
60
R=0
6 1 2 0
20
7 3 3 0 R=0
S=1 S=0 S=0 S=0
0≥
=
00200
600200
1060030
X
( ) 660=xf
150 120 80 50
100 3 5 7 11
130 1 4 6 3
170 5 8 12 7
150 120 80 50
100 3 0
20(-20)
5 0
80(+20)
7 -2 11 7 R=3
130 1 0
130
4 1 6 -1 3 1 R=5
170 5 -1
(+20)
8 0
40(-20)
12 0
80
7 0
50
R=0
S=-6 S=-8 S=-12 S=-7
- 79 -
80. ( ) ( ) ( ) ( ){ },1,1,2,1,2,3,1,3=V
{ } 2020,40min =
3 1
0
5 0
100(-80)
7 -2
(+80)
11 7 R=0
1 0
130
4 0 6 -2 3 0 R=-1
5 0
20
8 0
20(+80)
12 0
80(-80)
7 0
50
R=0
S=1 S=0 S=0 S=0
( ) ( ) ( ) ( ){ },3,3,2,3,2,1,3,1=V
{ } 8080,100min =
3 1
0
5 0
20
7 0
80
11 7 R=0
1 0
130
4 0 6 0 3 0 R=0
5 0
20
8 0
100
12 2
0
7 0
50
R=0
S=0 S=0 S=2 S=0
0≥
=
50010020
000130
080200
X
( ) 2040=xf
- 80 -
81. 1)
25 40 20 10
40 4 3 7 8
20 6 2 3 4
35 5 3 8 6
=
100250
02000
001525
X
( ) 340=xf
2)
220 310 200 250
300 8 5 4 6
500 12 11 9 13
180 10 15 18 14
=
000180
020026040
2500500
X
( ) 8690=xf
- 81 -
82. 3)
76 62 88 45 40
79 10 19 9 6 8
102 13 11 8 7 4
70 12 17 10 5 3
60 12 18 18 7 9
=
0150045
4030000
0040620
0048031
X
( ) 2659=xf
4)
85 75 60 50
105 4 16 10 14
65 10 18 12 20
55 6 4 14 18
45 8 6 8 12
=
250200
00550
56000
200085
X ( ) 2080=xf
- 82 -
83. 5)
120 280 130 270
100 6 8 3 7
300 9 10 11 4
150 5 7 9 10
250 12 13 8 9
=
0302200
0030120
2700300
010000
X
( ) 5590=xf
- 83 -
85. Trang
1
§ 1
§ 6
§ 9
15
§ 15
§ 20
26
41
§ 41
§ 44
47
59
§ 59
§ 61
§ 66
68
86
87
- 85 -
