Pembuktian Ketaksamaan Cauchy
Didefinisikan fungsi F : R → R sebagai berikut :
F(t) = (a1 – tb1)2 + (a2
– tb2)2 + ……..+ (an – tbn)2
, t є R
Jelas bahwa F(t) ≥ 0, untuk setiap t є R.
|
|
|
5
|
8
|
0
|
0
|
0
|
24/6
= 4
18/1
= 18
36/9 = 4
|
CB
|
VDB
|
B
|
X
|
Y
|
S1
|
S2
|
S3
|
|
0
|
S1
|
24
|
4
|
6
|
1
|
0
|
0
|
|
0
|
S2
|
18
|
2
|
1
|
0
|
1
|
0
|
|
0
|
S3
|
36
|
3
|
9*
|
0
|
0
|
1
|
|
Zj
– cj
|
0
|
-5
|
-8
|
0
|
0
|
0
|
|
|
|
5
|
8
|
0
|
0
|
0
|
0/2 = 4
|
CB
|
VDB
|
B
|
X
|
Y
|
S1
|
S2
|
S3
|
|
0
|
S1
|
0
|
2*
|
0
|
1
|
0
|
-2/3
|
|
0
|
S2
|
14
|
5/3
|
0
|
0
|
1
|
-1/9
|
|
8
|
Y
|
4
|
1/3
|
1
|
0
|
0
|
1/9
|
|
Zj
– cj
|
32
|
-7/3
|
0
|
0
|
0
|
8/9
|
|
|
|
5
|
8
|
0
|
0
|
0
|
CB
|
VDB
|
B
|
X
|
Y
|
S1
|
S2
|
S3
|
0
|
X
|
0
|
1
|
0
|
½
|
0
|
-1/3
|
0
|
S2
|
14
|
0
|
0
|
-5/6
|
1
|
4/9
|
8
|
Y
|
4
|
0
|
1
|
-1/6
|
0
|
2/9
|
Zj
– cj
|
32
|
0
|
0
|
7/6
|
0
|
1/9
|
|
|
|
5
|
8
|
0
|
0
|
0
|
CB
|
VDB
|
B
|
X
|
Y
|
S1
|
S2
|
S3
|
8
|
Y
|
4
|
2/3
|
1
|
1/6
|
0
|
0
|
0
|
S2
|
14
|
4/3
|
0
|
-1/6
|
1
|
0
|
0
|
S3
|
0
|
-3
|
0
|
-3/2
|
0
|
1
|
Zj
– cj
|
32
|
1/3
|
0
|
4/3
|
0
|
0
|
|
|
|
¾
|
-20
|
½
|
-6
|
0
|
0
|
0
|
CB
|
VDB
|
B
|
X1
|
X2
|
X3
|
X4
|
S1
|
S2
|
S3
|
0
|
S1
|
0
|
¼
|
-8
|
-1
|
9
|
1
|
0
|
0
|
0
|
S2
|
0
|
½
|
-12
|
-½
|
3
|
0
|
1
|
0
|
0
|
S3
|
1
|
0
|
0
|
1
|
0
|
0
|
0
|
1
|
Zj
- cj
|
0
|
-¾
|
20
|
-½
|
6
|
0
|
0
|
0
|
|
|
|
¾
|
-20
|
½
|
-6
|
0
|
0
|
0
|
CB
|
VDB
|
B
|
X1
|
X2
|
X3
|
X4
|
S1
|
S2
|
S3
|
¾
|
X1
|
0
|
1
|
-32
|
-4
|
36
|
4
|
0
|
0
|
0
|
S2
|
0
|
0
|
4
|
3/2
|
-15
|
-2
|
1
|
0
|
0
|
S3
|
1
|
0
|
0
|
1
|
0
|
0
|
0
|
1
|
Zj
- cj
|
0
|
0
|
-4
|
-7/2
|
33
|
3
|
0
|
0
|
|
|
|
¾
|
-20
|
½
|
-6
|
0
|
0
|
0
|
CB
|
VDB
|
B
|
X1
|
X2
|
X3
|
X4
|
S1
|
S2
|
S3
|
¾
|
X1
|
0
|
1
|
0
|
8
|
-2
|
-12
|
8
|
0
|
-20
|
X2
|
0
|
0
|
1
|
3/8
|
-15/4
|
-½
|
¼
|
0
|
0
|
S3
|
1
|
0
|
0
|
1
|
0
|
0
|
0
|
1
|
Zj
- cj
|
0
|
0
|
0
|
-2
|
18
|
1
|
1
|
0
|
|
|
|
¾
|
-20
|
½
|
-6
|
0
|
0
|
0
|
CB
|
VDB
|
B
|
X1
|
X2
|
X3
|
X4
|
S1
|
S2
|
S3
|
½
|
X3
|
0
|
1/8
|
0
|
1
|
-21/4
|
-3/2
|
1
|
0
|
-20
|
X2
|
0
|
-3/64
|
1
|
0
|
3/16
|
1/16
|
-1/8
|
0
|
0
|
S3
|
1
|
-1/8
|
0
|
0
|
21/2
|
3/2
|
-1
|
1
|
Zj
- cj
|
0
|
¼
|
0
|
0
|
-3
|
-2
|
3
|
0
|
|
|
|
¾
|
-20
|
½
|
-6
|
0
|
0
|
0
|
CB
|
VDB
|
B
|
X1
|
X2
|
X3
|
X4
|
S1
|
S2
|
S3
|
½
|
X3
|
0
|
-5/2
|
56
|
1
|
0
|
2
|
-6
|
0
|
-6
|
X4
|
0
|
-¼
|
16/3
|
0
|
1
|
1/3
|
2/3
|
0
|
0
|
S3
|
1
|
5/2
|
-56
|
0
|
0
|
-2
|
6
|
1
|
Zj
- cj
|
0
|
-½
|
16
|
0
|
0
|
-1
|
1
|
0
|
|
|
|
¾
|
-20
|
½
|
-6
|
0
|
0
|
0
|
CB
|
VDB
|
B
|
X1
|
X2
|
X3
|
X4
|
S1
|
S2
|
S3
|
0
|
S1
|
0
|
-5/4
|
28
|
½
|
0
|
1
|
-3
|
0
|
-6
|
X4
|
0
|
1/6
|
-4
|
-1/6
|
1
|
0
|
1/3
|
0
|
0
|
S3
|
1
|
0
|
0
|
1
|
0
|
0
|
1/3
|
1
|
Zj
- cj
|
0
|
-7/4
|
44
|
½
|
0
|
0
|
-2
|
0
|
|
|
|
¾
|
-20
|
½
|
-6
|
0
|
0
|
0
|
CB
|
VDB
|
B
|
X1
|
X2
|
X3
|
X4
|
S1
|
S2
|
S3
|
0
|
S1
|
0
|
¼
|
-8
|
-1
|
9
|
1
|
0
|
0
|
0
|
S2
|
0
|
½
|
-12
|
-½
|
3
|
0
|
1
|
0
|
0
|
S3
|
1
|
0
|
0
|
1
|
0
|
0
|
0
|
1
|
Zj
- cj
|
0
|
-¾
|
20
|
-½
|
6
|
0
|
0
|
0
|
|
|
|
¾
|
-20
|
½
|
-6
|
0
|
0
|
0
|
CB
|
VDB
|
B
|
X1
|
X2
|
X3
|
X4
|
S1
|
S2
|
S3
|
0
|
S1
|
0
|
0
|
-2
|
-3/4
|
15/2
|
1
|
-1/2
|
0
|
¾
|
X1
|
0
|
1
|
-24
|
-1
|
6
|
0
|
2
|
0
|
0
|
S3
|
1
|
0
|
0
|
1
|
0
|
0
|
0
|
1
|
Zj
- cj
|
0
|
0
|
2
|
-5/4
|
21/2
|
0
|
3/2
|
0
|
|
|
|
¾
|
-20
|
½
|
-6
|
0
|
0
|
0
|
CB
|
VDB
|
B
|
X1
|
X2
|
X3
|
X4
|
S1
|
S2
|
S3
|
0
|
S1
|
¾
|
0
|
-2
|
0
|
15/2
|
1
|
-1/2
|
-3/4
|
¾
|
X1
|
1
|
1
|
-24
|
0
|
6
|
0
|
2
|
1
|
½
|
X3
|
1
|
0
|
0
|
1
|
0
|
0
|
0
|
1
|
Zj
- cj
|
5/4
|
0
|
2
|
0
|
21/2
|
0
|
3/2
|
5/4
|