Post on 05-Jul-2015
transcript
Inequality Techniques
Inequality Techniques0proveeasier tobecan it ,prove To yxyx
Inequality Techniques0proveeasier tobecan it ,prove To yxyx
2
Prove 1995 e.g.22 qppqi
Inequality Techniques0proveeasier tobecan it ,prove To yxyx
2
Prove 1995 e.g.22 qppqi
pqqp
2
22
Inequality Techniques0proveeasier tobecan it ,prove To yxyx
2
Prove 1995 e.g.22 qppqi
pqqp
2
22
22 22 qpqp
Inequality Techniques0proveeasier tobecan it ,prove To yxyx
2
Prove 1995 e.g.22 qppqi
pqqp
2
22
22 22 qpqp
2
2qp
Inequality Techniques0proveeasier tobecan it ,prove To yxyx
2
Prove 1995 e.g.22 qppqi
pqqp
2
22
22 22 qpqp
2
2qp
0
Inequality Techniques0proveeasier tobecan it ,prove To yxyx
2
Prove 1995 e.g.22 qppqi
pqqp
2
22
22 22 qpqp
2
2qp
0
pqqp
2
22
acbcabcbaii 222 Prove a) 1994
acbcabcbaii 222 Prove a) 1994 02 ba
acbcabcbaii 222 Prove a) 1994 02 ba
02 22 baba
acbcabcbaii 222 Prove a) 1994 02 ba
02 22 babaabba 222
acbcabcbaii 222 Prove a) 1994 02 ba
02 22 babaabba 222
bccbacca
22
22
22
acbcabcbaii 222 Prove a) 1994 02 ba
02 22 babaabba 222
bccbacca
22
22
22
bcacabcba 222222 222
acbcabcbaii 222 Prove a) 1994 02 ba
02 22 babaabba 222
bccbacca
22
22
22
bcacabcba 222222 222 bcacabcba 222
acbcabcbaii 222 Prove a) 1994 02 ba
02 22 babaabba 222
bccbacca
22
22
22
bcacabcba 222222 222 bcacabcba 222
31 prove ,1 If b) bcacabcba
acbcabcbaii 222 Prove a) 1994 02 ba
02 22 babaabba 222
bccbacca
22
22
22
bcacabcba 222222 222 bcacabcba 222
31 prove ,1 If b) bcacabcba
bcacabcbacba 22222
acbcabcbaii 222 Prove a) 1994 02 ba
02 22 babaabba 222
bccbacca
22
22
22
bcacabcba 222222 222 bcacabcba 222
31 prove ,1 If b) bcacabcba
bcacabcbacba 22222
bcacabbcacabcba 22
acbcabcbaii 222 Prove a) 1994 02 ba
02 22 babaabba 222
bccbacca
22
22
22
bcacabcba 222222 222 bcacabcba 222
31 prove ,1 If b) bcacabcba
bcacabcbacba 22222
bcacabbcacabcba 22
23 cbabcacab
acbcabcbaii 222 Prove a) 1994 02 ba
02 22 babaabba 222
bccbacca
22
22
22
bcacabcba 222222 222 bcacabcba 222
31 prove ,1 If b) bcacabcba
bcacabcbacba 22222
bcacabbcacabcba 22
23 cbabcacab
3113
bcacab
bcacab
3
31 Prove c) abccba
3
31 Prove c) abccba
bcacabcba 222
3
31 Prove c) abccba
bcacabcba 222
0222 bcacabcba
3
31 Prove c) abccba
bcacabcba 222
0222 bcacabcba 0222 bcacabcbacba
3
31 Prove c) abccba
bcacabcba 222
0222 bcacabcba 0222 bcacabcbacba
022322
2223222223
bcacabcccbcacbabcabbcbbaabccabaacaba
3
31 Prove c) abccba
bcacabcba 222
0222 bcacabcba 0222 bcacabcbacba
022322
2223222223
bcacabcccbcacbabcabbcbbaabccabaacaba
03333 abccba
3
31 Prove c) abccba
bcacabcba 222
0222 bcacabcba 0222 bcacabcbacba
022322
2223222223
bcacabcccbcacbabcabbcbbaabccabaacaba
03333 abccba
abccba 333
31
3
31 Prove c) abccba
bcacabcba 222
0222 bcacabcba 0222 bcacabcbacba
022322
2223222223
bcacabcccbcacbabcabbcbbaabccabaacaba
03333 abccba
abccba 333
31
31
31
31
,,let ccbbaa
3
31 Prove c) abccba
bcacabcba 222
0222 bcacabcba 0222 bcacabcbacba
022322
2223222223
bcacabcccbcacbabcabbcbbaabccabaacaba
03333 abccba
abccba 333
31
31
31
31
,,let ccbbaa 3
131
31
31 cbacba
3
31 Prove c) abccba
bcacabcba 222
0222 bcacabcba 0222 bcacabcbacba
022322
2223222223
bcacabcccbcacbabcabbcbbaabccabaacaba
03333 abccba
abccba 333
31
31
31
31
,,let ccbbaa 3
131
31
31 cbacba
3
31 abccba
nn
n aaan
aaa 21
21
MeanGeometricMeanArithmetic
nn
n aaan
aaa 21
21
MeanGeometricMeanArithmetic
1prove,8111 Suppose d) xyzzyx
nn
n aaan
aaa 21
21
MeanGeometricMeanArithmetic
1prove,8111 Suppose d) xyzzyx
818111
xyzyzxzzxyyxzyx
nn
n aaan
aaa 21
21
MeanGeometricMeanArithmetic
1prove,8111 Suppose d) xyzzyx
818111
xyzyzxzzxyyxzyx
3
31 xyzzyx
nn
n aaan
aaa 21
21
MeanGeometricMeanArithmetic
1prove,8111 Suppose d) xyzzyx
818111
xyzyzxzzxyyxzyx
3
31 xyzzyx
33 xyzzyx
nn
n aaan
aaa 21
21
MeanGeometricMeanArithmetic
1prove,8111 Suppose d) xyzzyx
818111
xyzyzxzzxyyxzyx
3
31 xyzzyx
33 xyzzyx
33 xzyzxyxzyzxy
nn
n aaan
aaa 21
21
MeanGeometricMeanArithmetic
1prove,8111 Suppose d) xyzzyx
818111
xyzyzxzzxyyxzyx
3
31 xyzzyx
33 xyzzyx
33 xzyzxyxzyzxy
3 2223 zyxxzyzxy
nn
n aaan
aaa 21
21
MeanGeometricMeanArithmetic
1prove,8111 Suppose d) xyzzyx
818111
xyzyzxzzxyyxzyx
3
31 xyzzyx
33 xyzzyx
33 xzyzxyxzyzxy
3 2223 zyxxzyzxy
233 xyzxzyzxy
81 xyzyzxzxyzyx
81 xyzyzxzxyzyx 8331 233 xyzxyzxyz
81 xyzyzxzxyzyx 8331 233 xyzxyzxyz
8331 33233 xyzxyzxyz
81 xyzyzxzxyzyx 8331 233 xyzxyzxyz
8331 33233 xyzxyzxyz
81 33 xyz
81 xyzyzxzxyzyx 8331 233 xyzxyzxyz
8331 33233 xyzxyzxyz
81 33 xyz
21 3 xyz
81 xyzyzxzxyzyx 8331 233 xyzxyzxyz
8331 33233 xyzxyzxyz
81 33 xyz
21 3 xyz
13 xyz
81 xyzyzxzxyzyx 8331 233 xyzxyzxyz
8331 33233 xyzxyzxyz
81 33 xyz
21 3 xyz
13 xyz
1xyz
81 xyzyzxzxyzyx 8331 233 xyzxyzxyz
8331 33233 xyzxyzxyz
81 33 xyz
21 3 xyz
13 xyz
1xyz
Inequalities Sheet
Exercise 10D
Note: Cambridge 8H (Book 1); 28