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