X2 t08 04 inequality techniques (2013)

Types of Proof

Types of Proof 1. Deductive Proof

Start with known facts and deduce what you are trying to prove.



2. Inductive Proof

Assume what you are trying to prove and induce a solution.



2. Inductive Proof

Assume what you are trying to prove and induce a solution.

3. Proof by Contradiction

Assume the opposite of what you are trying to prove and create a

contradiction.

Contradiction means the original assumption is incorrect, therefore

the opposite must be true.

Inequality Techniques

Inequality Techniques 0 prove easier to becan it , prove To yxyx


2

Prove 1995 e.g.22 qp

pqi


2


pqi

pqqp

2

22


2


pqi

pqqp

2

22

2

2 22 qpqp


2


pqi

pqqp

2

22

2

2 22 qpqp

2

2qp

0


2


pqi

pqqp

2

22

2

2 22 qpqp

2

2qp

0

pqqp

2

22


2


pqi

pqqp

2

22

2

2 22 qpqp

2

2qp

0

pqqp

2

22

OR 2 2

Assume 2

p qpq


2


pqi

pqqp

2

22

2

2 22 qpqp

2

2qp

0

pqqp

2

22

OR 2 2

Assume 2

p qpq

2 22pq p q

2 2

2

0 2

0 ( )

p pq q

p q


2


pqi

pqqp

2

22

2

2 22 qpqp

2

2qp

0

pqqp

2

22

OR 2 2

Assume 2

p qpq

2 22pq p q

2 2

2

0 2

0 ( )

p pq q

p q

2But ( ) 0p q


2


pqi

pqqp

2

22

2

2 22 qpqp

2

2qp

0

pqqp

2

22

OR 2 2

Assume 2

p qpq

2 22pq p q

2 2

2

0 2

0 ( )

p pq q

p q

2But ( ) 0p q 2 2

2

p qpq

acbcabcbaii 222 Prove a) 1994


02ba


02ba

02 22 baba


02ba

02 22 baba

abba 222


02ba

02 22 baba

abba 222

bccb

acca

2

2

22

22


02ba

02 22 baba

abba 222

bccb

acca

2

2

22

22

bcacabcba 222222 222


02ba

02 22 baba

abba 222

bccb

acca

2

2

22

22


bcacabcba 222


02ba

02 22 baba

abba 222

bccb

acca

2

2

22

22


bcacabcba 222

3

1 prove ,1 If b) bcacabcba


02ba

02 22 baba

abba 222

bccb

acca

2

2

22

22


bcacabcba 222

3


bcacabcbacba 22222


02ba

02 22 baba

abba 222

bccb

acca

2

2

22

22


bcacabcba 222

3


bcacabcbacba 22222

bcacabbcacabcba 22


02ba

02 22 baba

abba 222

bccb

acca

2

2

22

22


bcacabcba 222

3


bcacabcbacba 22222

bcacabbcacabcba 22

23 cbabcacab


02ba

02 22 baba

abba 222

bccb

acca

2

2

22

22


bcacabcba 222

3


bcacabcbacba 22222

bcacabbcacabcba 22

23 cbabcacab

3

1

13

bcacab

bcacab

3

3

1 Prove c) abccba

3

3

1 Prove c) abccba

bcacabcba 222

3

3

1 Prove c) abccba

bcacabcba 222

0222 bcacabcba

3

3

1 Prove c) abccba

bcacabcba 222

0222 bcacabcba

0222 bcacabcbacba

3

3

1 Prove c) abccba

bcacabcba 222

0222 bcacabcba

0222 bcacabcbacba

022322

2223222223

bcacabcccbca

cbabcabbcbbaabccabaacaba

3

3

1 Prove c) abccba

bcacabcba 222

0222 bcacabcba

0222 bcacabcbacba

022322

2223222223

bcacabcccbca


03333 abccba

3

3

1 Prove c) abccba

bcacabcba 222

0222 bcacabcba

0222 bcacabcbacba

022322

2223222223

bcacabcccbca


03333 abccba

abccba 333

3

1

3

3

1 Prove c) abccba

bcacabcba 222

0222 bcacabcba

0222 bcacabcbacba

022322

2223222223

bcacabcccbca


03333 abccba

abccba 333

3

1

3

1

3

1

3

1

,,let ccbbaa

3

3

1 Prove c) abccba

bcacabcba 222

0222 bcacabcba

0222 bcacabcbacba

022322

2223222223

bcacabcccbca


03333 abccba

abccba 333

3

1

3

1

3

1

3

1

,,let ccbbaa

3

1

3

1

3

1

3

1cbacba

3

3

1 Prove c) abccba

bcacabcba 222

0222 bcacabcba

0222 bcacabcbacba

022322

2223222223

bcacabcccbca


03333 abccba

abccba 333

3

1

3

1

3

1

3

1

,,let ccbbaa

3

1

3

1

3

1

3

1cbacba

3

3

1abccba

nn

n aaan

aaa

21

21

Mean GeometricMean Arithmetic

nn

n aaan

aaa

21

21


1 prove ,8111 Suppose d) xyzzyx

nn

n aaan

aaa

21

21



81

8111

xyzyzxzzxyyx

zyx

nn

n aaan

aaa

21

21



81

8111

xyzyzxzzxyyx

zyx

3

3

1xyzzyx AM GM

nn

n aaan

aaa

21

21



81

8111

xyzyzxzzxyyx

zyx

3

3

1xyzzyx

33 xyzzyx

AM GM

nn

n aaan

aaa

21

21



81

8111

xyzyzxzzxyyx

zyx

3

3

1xyzzyx

33 xyzzyx

33 xzyzxyxzyzxy

AM GM

nn

n aaan

aaa

21

21



81

8111

xyzyzxzzxyyx

zyx

3

3

1xyzzyx

33 xyzzyx

33 xzyzxyxzyzxy

3 2223 zyxxzyzxy

AM GM

nn

n aaan

aaa

21

21



81

8111

xyzyzxzzxyyx

zyx

3

3

1xyzzyx

33 xyzzyx

33 xzyzxyxzyzxy

3 2223 zyxxzyzxy

233 xyzxzyzxy

AM GM

81 xyzyzxzxyzyx

81 xyzyzxzxyzyx

83312

33 xyzxyzxyz

81 xyzyzxzxyzyx

83312

33 xyzxyzxyz

83313

32

33 xyzxyzxyz

81 xyzyzxzxyzyx

83312

33 xyzxyzxyz

83313

32

33 xyzxyzxyz

813

3 xyz

81 xyzyzxzxyzyx

83312

33 xyzxyzxyz

83313

32

33 xyzxyzxyz

813

3 xyz

21 3 xyz

81 xyzyzxzxyzyx

83312

33 xyzxyzxyz

83313

32

33 xyzxyzxyz

813

3 xyz

21 3 xyz

13 xyz

81 xyzyzxzxyzyx

83312

33 xyzxyzxyz

83313

32

33 xyzxyzxyz

813

3 xyz

21 3 xyz

13 xyz

1xyz

OR 1 2x x AM GM

OR 1 2x x

1 2y y

1 2z z

AM GM

OR 1 2x x

1 2y y

1 2z z

(1 )(1 ) 1 2 2 2

8

x y z x y z

xyz

AM GM

OR 1 2x x

1 2y y

1 2z z

(1 )(1 ) 1 2 2 2

8

x y z x y z

xyz

8 8 xyz

AM GM

OR 1 2x x

1 2y y

1 2z z

(1 )(1 ) 1 2 2 2

8

x y z x y z

xyz

8 8 xyz

1 1

1

1

xyz

xyz

xyz

AM GM

9 2 2 2 1 1 1

Prove iiia b c a b b c a c a b c

9 2 2 2 1 1 1


1 1 1

2 2

4

a b aba b ab

9 2 2 2 1 1 1


1 1 1

2 2

4

a b aba b ab

1 1 4

a b a b

9 2 2 2 1 1 1


1 1 1

2 2

4

a b aba b ab

1 1 4

a b a b

1 1 4

1 1 4

b c b c

a c a c

9 2 2 2 1 1 1


1 1 1

2 2

4

a b aba b ab

1 1 4

a b a b

1 1 4

1 1 4

b c b c

a c a c

2 2 2 4 4 4

a b c a b b c a c

9 2 2 2 1 1 1


1 1 1

2 2

4

a b aba b ab

1 1 4

a b a b

1 1 4

1 1 4

b c b c

a c a c

2 2 2 4 4 4

a b c a b b c a c

1 1 1 2 2 2

a b c a b b c a c

9 2 2 2 1 1 1


1 1 1

2 2

4

a b aba b ab

1 1 4

a b a b

1 1 4

1 1 4

b c b c

a c a c

2 2 2 4 4 4

a b c a b b c a c

1 1 1 2 2 2

a b c a b b c a c

3 31 1 1 1

3 3

9

a b c abca b c abc

9 2 2 2 1 1 1


1 1 1

2 2

4

a b aba b ab

1 1 4

a b a b

1 1 4

1 1 4

b c b c

a c a c

2 2 2 4 4 4

a b c a b b c a c

1 1 1 2 2 2

a b c a b b c a c

3 31 1 1 1

3 3

9

a b c abca b c abc

1 1 1 9

a b c a b c

9 2 2 2 1 1 1


1 1 1

2 2

4

a b aba b ab

1 1 4

a b a b

1 1 4

1 1 4

b c b c

a c a c

2 2 2 4 4 4

a b c a b b c a c

1 1 1 2 2 2

a b c a b b c a c

3 31 1 1 1

3 3

9

a b c abca b c abc

1 1 1 9

a b c a b c

1 1 1 9

2

2 2 2 9

a b b c a c a b c

a b b c a c a b c

9 2 2 2 1 1 1


1 1 1

2 2

4

a b aba b ab

1 1 4

a b a b

1 1 4

1 1 4

b c b c

a c a c

2 2 2 4 4 4

a b c a b b c a c

1 1 1 2 2 2

a b c a b b c a c

3 31 1 1 1

3 3

9

a b c abca b c abc

1 1 1 9

a b c a b c

1 1 1 9

2

2 2 2 9

a b b c a c a b c

a b b c a c a b c

9 2 2 2 1 1 1

a b c a b b c a c a b c

9 2 2 2 1 1 1


1 1 1

2 2

4

a b aba b ab

1 1 4

a b a b

1 1 4

1 1 4

b c b c

a c a c

2 2 2 4 4 4

a b c a b b c a c

1 1 1 2 2 2

a b c a b b c a c

3 31 1 1 1

3 3

9

a b c abca b c abc

1 1 1 9

a b c a b c

1 1 1 9

2

2 2 2 9

a b b c a c a b c

a b b c a c a b c

9 2 2 2 1 1 1

a b c a b b c a c a b c

Inequalities Sheet

Exercise 10D

Note: Cambridge 8H (Book 1); 28

X2 t08 04 inequality techniques (2013)

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