Inequalities
Advanced Level Pure Mathematics
Advanced Level Pure Mathematics
6
Algebra
Chapter 6
Inequalities
Fundamental Concepts of Inequalities and
Methods of Proving Inequalities
2
6.4
Arithmetic Mean and Geometric Mean
7
6.5
Cauchy-Schwarz Inequality
18
6.6
Absolute Values
22
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Prepared by K. F. Ngai
Inequalities
Advanced Level Pure Mathematics
Fundamental Concepts of Inequalities and
Methods of Proving Inequalities Algebraic Inequalities
Example 1
1.
2.
3.
4.
For any real number a , a 2  0. Equality holds iff a  0 .
If a  b and b  c, then a  c.
If a  b , then a  c  b  c, c  R.
If a  b , then ac  bc, c  0 and ac  bc, c  0 .
5.
If a  b  0, then a p  b p for any p  0. Equality holds iff q  0.
(a) Prove that for any x, y  0,
2( x 2  y 2 )  x  y  2 xy.
(b) Hence or otherwise, deduce that if a, b, c  0, then
a 2  b 2  b 2  c 2  c 2  a 2  2 ( a  b  c)
(i)
(ii) (a  b)(b  c)(c  a)  8abc
Solution
(a) (i)
x  y  2 xy  ( x  y ) 2  0

x  y  2 xy
(ii) 2( x 2  y 2 )  ( x  y) 2
(b) (i)
=
=

2( x 2  y 2 )  ( x  y) 2

2(x 2  y 2 )  x  y
By (a), we have
2 x 2  2 y 2  x 2  2 xy  y 2
( x  y) 2  0
( x, y  0)
a2  b2 
c2  a2 
1
2
1
2
( a  b) , b 2  c 2 
1
2
(b  c) and
(c  a)
Adding up the inequalities, we have
a2  b2  b2  c2  c2  a2


(ii) By (a)
we have a  b  2 ab
1
2
2
(a  b  b  c  c  a)
2
(a  b  c)  2 (a  b  c)
( 0) ,
b  c  2 bc
( 0)
c  a  2 ca
( 0)
Multiplying the inequalities, we have (a  b)(b  c)(c  a)  8 ab bc ca
 8abc
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Prepared by K. F. Ngai
Inequalities
Advanced Level Pure Mathematics
Example 2
Given that a, b, c  R. Prove that a  b  c  ab  bc  ca
Example 3
Prove that x, y  0,
2
2
2
1 1
4
.
 
x y x y
Equality sign holds iff x  y.
Example 4
Show that a, b  R,
(a)
a2  b2 
1
( a  b) 2
2
(b) a (1  a ) 
Example 5
Show that sin x  x, x  (0,  ) .
Example 6
Show that 3x 2  5xy  3 y 2  0, x, y  R .
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Prepared by K. F. Ngai
Inequalities
Example 6.9
Advanced Level Pure Mathematics
If a, b, c denote the lengths of the sides of a triangle, prove that
(a  b  c)(b  c  a )
4
bc
Solution
Example 6.10
Let a, m, n be positive real numbers not equal to 1 and m  n . Prove that
a m  a m  a n  a n
Solution
Example 6.11
Let a, b and c be real numbers such that a  b  c  1 .Prove that
a2  b2  c2 
1
3
When does equality hold?
Solution
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Prepared by K. F. Ngai
Inequalities
Example 6.12
Advanced Level Pure Mathematics
Let a, b be positive real number. Prove that
a abb  a bb a
Solution
Example 6.18
Show that for any positive integer k ,
1
k 1  k 
2 k
Hence deduce that for any positive integer n ,
1
1
1
1


 2( n  1  1)
2
3
n
Solution
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Prepared by K. F. Ngai
Inequalities
Example 6.14
Advanced Level Pure Mathematics
Let n be a positive integer. Prove that
2n  1
2n  1

2n
2n  1
Solution
Example 6.19
Show that for any positive integer k ,
2k  1
3k  2

2k
3k  1
Hence deduce that for any positive integer n ,
1 3 5
2n  1
1
( )( )( ) (
)
2 4 6
2n
3n  1
Solution
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Prepared by K. F. Ngai
Inequalities
6.4
Advanced Level Pure Mathematics
Arithmetic Mean and Geometric Mean
A.M.  G.M.
Let x and y be two positive real numbers. We have
Theorem 6-3
x y
 xy
2
and equality holds if and only if x  y
Proof
Equality holds if and only if
x y 0
i.e.
x y
xy
Definition
Let a1 , a2 ,, an be n non-negative real numbers.
A.M.=
a1  a2    an
and
n
G.M.=
n
a1 a 2  a n
are respectively the Arithmetic Mean (A.M.) and Geometric Mean (G.M.) of the given numbers.
Theorem
For any n non-negative numbers a1 , a2 ,, an A.M .  G.M .
a1  a2    an n
 a1 a 2  a n
n
The equality holds iff a1  a2    an
i.e.
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Prepared by K. F. Ngai
Inequalities
Example 7
Advanced Level Pure Mathematics
By considering the A.M. and G.M. of a suitable set of numbers,
prove the following. ( n  N ).
(a)
2 n  1  n 2 n 1 ,
1,2,4,,2 
(b)
(n  1)!2  (n  1) n 1 ,
 1 1
1 
,
, ,


n(n  1) 
1  2 2  3
n 1
Solution
Example 8
Give that a, b, c  0. Prove the following.
(a)
a b c
  3
b c a
(b) a 2 (1  b 2 )  b 2 (1  c 2 )  c 2 (1  a 2 )  6abc
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Prepared by K. F. Ngai
Inequalities
Advanced Level Pure Mathematics
(c)
Example 9
a  b  c  ab  bc  ca
Let a, b, c be distinct positive numbers, show that
2
2
2
9



.
ab bc ca abc
Solution
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Prepared by K. F. Ngai
Inequalities
Advanced Level Pure Mathematics
AL-01I-14
(a) If a, b are two real numbers such that a  1  b , show that a  b  ab  1 and the equality holds if
and only if a  1 or b  1 .
(b) Show by induction that if x1 , x2 ,, xn are n (n  2) positive real numbers such that x1 , x2 ,, xn ,
then x1  x2    xn  n and the equality holds if and only if x1  x2    xn  1
(c) Let a1 , a2 ,, an be n (n  2) positive real numbers. Using (b) or otherwise, show that
a1  a2    an

n
n
a1 a 2  a n
and the equality holds if and only if a1  a2    an .
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Prepared by K. F. Ngai
Inequalities
Advanced Level Pure Mathematics
Example 10 For all non-negative integer n, prove that
(a)
(
n 1 n
)  n! ,
2
1
(b) 1 
for any n
1
n
(n!) n  (( n  1)!) n 1
n 1
Solution
Example 11
(a) Let a1 , a2 ,, an be n positive numbers. Prove that
 n  n 1
2
 a i    a   n .
 i 1   i 1 i 
Equality holds iff a1  a2    an .
(b) Hence, or otherwise, prove that  n  N ,
(i)
1
1 1
1
2n
  
2 3
n n 1
(ii) (2 n  1) 2  n 2  2 n1
Solution
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Prepared by K. F. Ngai
Inequalities
Advanced Level Pure Mathematics
Example 12 If x  1, show that x
2 n 1
 1  (2n  1)( x  1) x
n
Solution
2
3

Example 13 By using A.M .  G.M . , show that  k    12 .
k

3
Hence deduce that the roots of the quadratic equation x 2  (k  ) x  2  0 are real and
k
distinct.
Solution
AL-92I-7
C rn 1
(a) Prove that r  ﹐where n , r are positive integers and n  r ．
r!
n
(b) If a1 , a2 ,  , an are positive numbers and
s  a1  a2    an ﹐using " A.M .  G.M " and (a)﹐or otherwise ﹐prove that
s2 s3
sn
1  a1 1  a2 1  an   1  s      .
2! 3!
n!
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Prepared by K. F. Ngai
Inequalities
Advanced Level Pure Mathematics
Example 14 Let a, b, c, d be non-negative numbers, prove the following
(a)
(a  b  c)( a 2  b 2  c 2 )  9abc
(b) (a  b  c  d )( a 3  b 3  c 3  d 3 )  16abcd
Solution
Example 15 Let a, b, c be positive numbers, prove the following:
1 1 1 1  1 1
(b) a    b    c    6
b c c a a b
c
a
b
3



(d)
ab bc ca 2
(a)
a b c
  3
b c a
(c)
1 1 1
1
1
1
  


a b c
ab
bc
ca
(e)
1 1 1
2
2
2
  


a b c ab bc ca
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Prepared by K. F. Ngai
Inequalities
Advanced Level Pure Mathematics
n
Note:
n
 i   n  1  2    n  2 (n  1)
i 1
n
2

n
(n  1)( 2n  1)
6
n2
 n  4 (n  1) 2
Example 16 Show that if n is a positive integer greater that 1.
3
2
(a)
(n!) n 
(n  1)( 2n  1)
6
3
(b) (n!) n 
page
n(n  1) 2
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Prepared by K. F. Ngai
Inequalities
Advanced Level Pure Mathematics
Example 17 If a, b are positive numbers, such that a  b , prove that
(ma  nb) mn  (m  n) mn a m b n
if
(i) m and n are positive numbers.
(ii) m and n are positive fractions.
Solution
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Prepared by K. F. Ngai
Inequalities
Advanced Level Pure Mathematics
Example 18 Let n be a positive integer greater than 1 , prove the following:
(a) 1  3  5(2n  1)  n n
(b) 2  4  6 2n  (n  1) n
n
(c)
1
2
(
n

1
)(
2
n

1
)
 6
  n!
(e)
3n  1
 3 n 1
2n
n
 4n 2  1  2
  1  3  5 (2n  1)
(d) 
 3 
Solution
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Prepared by K. F. Ngai
Inequalities
Advanced Level Pure Mathematics
Example 19 If a1 , a2 , p, q, r, are positive integers. Show that
(a)
(b)
a1
a1
p q
 a2
p q
p  q  r 
 a1 a 2  a1 a 2
p
 a2
2
q
p  q  r 
p
a1  a 2
a  a2
a  a2
)( 1
)( 1
)
2
2
2
p
(
p
q
q
r
r
a1  a 2
 a  a2 

 1
 ,n Z
2
 2 
n
(c) Hence show
q
n
n
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Inequalities
6.5
Theorem
Advanced Level Pure Mathematics
Cauchy-Schwarz Inequality
(Cauchy-Schwarz Inequality)
If a1 , a2 ,, an and b1 , b2 ,, bn are two sets of real numbers, then
2
 n 2  n 2   n

  a r   br     a r br  .
 r 1
 r  1   r  1

Equality holds if and only if
Proof
a
a1 a2

 n .
b1 b2
bn
Define the quadratic polynomial f (x) by
f(x) 
n
 (a
r1
n
i.e.
 (a
r1
r
2
r
x  br ) 2  0 , which is non-negative for all real values x.
x 2  2a r br x  br )  0
2
 n 2 2
 n
  n 2
  ar  x  2  ar br  x    br   0
 r 1 
 r 1
  r 1 
 f ( x)  0    0
2
 n

 n 2  n 2
i.e. 2 ar br   4 ar   br   0
 r 1

r 1  r 1 
2

 n

 n 2  n 2
a
b


r
r


  a r    br 
r  1

r  1  r  1 
n
Equality holds if and only if
 (a x  b
r1
r
r
)2  0
if and only if ar x  br  0 for any r
br
, r
ar
if and only if
x
if and only if
a
a1 a2

 n
b1 b2
bn
1
Example 20 Let a, b, c be real numbers, prove that a 2  b 2  c 2  (a  b  c) 2
3
Solution
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Prepared by K. F. Ngai
Inequalities
Advanced Level Pure Mathematics
Example 21 Let a, b, c are non-zero real numbers, show that (a 2  b 2  c 2 )(
1
1
1
 2  2)  9.
2
a
b
c
Solution
Example 22 Let a1 , a2 ,, an be real numbers..
Show that
1 n
1 n 2
a

ak .
 k n
n k 1
k 1
Solution
Example 23 Let n be a positive integer greater than 1. By using the Cauchy-Schwarz inequality,
show that
n(2 n  1)  C1n  C2n    Cnn
Solution
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Prepared by K. F. Ngai
Inequalities
Advanced Level Pure Mathematics
Example 24 Given that a, b, c  0 and a  b  c  3. Prove that
Solution
(a) a 2  b 2  c 2  3
(b) a 3  b 3  c 3  3
ab  bc  ca  3
(c)
Apply the Cauchy-Schwarz Inequality:
(a) Consider the numbers a, b, c,1,1,1. We have
Example 25 For any positive number a1 , a2 ,, an
(a) Prove (a1  a 2    a n )(
1
1
1

   )  n2
a1 a 2
an
Equality holds iff a1  a2    an .
Hence, or otherwise, prove that n  N ,
(i)
1
1 1
1
2n
  
2 3
n n 1
(ii) (2 n  1) 2  n 2  2 n1
(b) Prove that for any n positive numbers x1 , x2 ,, xn , the following inequality holds.
x  x2    xn
 x1  x2    xn 

  1
n
n


2
2
2
2
Hence, or otherwise, if x, y and z are three positive real numbers and x  y  z  6,
prove that x 2  y 2  z 2  12 .
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Inequalities
Advanced Level Pure Mathematics
Example 26 Prove that for any positive numbers a, b, c, d ;
a b c d
2
When does the equality hold?
abcd 
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Inequalities
6.6
Advanced Level Pure Mathematics
Absolute Values
Definition
The absolute value of a real number x , denoted by x ,
 x if
is defined by x  
 x if
x0
x0
Properties
(1)
x 0 x 0
(2)
x 0 x 0
(3)
x   x and a  b  b  a
(4)
x  a  x  a
(5)
b  a  a  b  a
if  a  b  a, a  0 then
Conversely,
(6)
b  a  b  a Or b  a
if b  a Or b  a, then
Conversely,
Remark:
Theorem
b  a.
b a
x  a  x 2  a 2 unless a  0
For any two real numbers a and b , the following hold:
(a) Triangle inequality. i.e.
(b)
(c)
ab  a  b
a b  a  b
a  b
 ab
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Prepared by K. F. Ngai
Inequalities
Theorem
Advanced Level Pure Mathematics
For any two real numbers a and b , the following hold:
(a)
ab  a b
(b) For any positive integer n , a n  a .
n
(c)
a
a
 ,b  0
b
b
Example 27 Solve 2  x  7  9 and 0  x  2  8
Solution
Consider x  7  9
Consider x  2  8
9  x 7  9
 16  x  2
8  x  2  8
 10  x  6
Consider x  7  2
Consider x  2  0
x  5 or x  9
The Solution:
 5  x  2 and 16  x  9


Example 28
For all Real number except x  2
 5  x  2 and 10  x  9 except x  2 .
x 2  3x  1  3
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Inequalities
AL-90I-6
Advanced Level Pure Mathematics
Solve the inequality x  1  x  2  2
Solution
AL-93I-7
Find all ( x, y ) in R 2 satisfying the following two conditions:
 2x  1  y  1

 y  x3
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Prepared by K. F. Ngai
Inequalities
Advanced Level Pure Mathematics
AL-90I-12
(a) Let p  1 and f x   x p  px for all x  0 ．
(i) Find the absolute minimum of f x  on the interval (0 ,  ) ．
(b) (i)
(ii) Deduce that x p   px  1 for all x  0 ．
Let  ,  ,  and  be positive numbers such that
1
1
 1 and     1

By taking x   and  respectively﹐prove that﹐for p  1 ﹐  p 1 p   p 1 p  1 ﹐
where the equality holds if and only if     1 ．
(ii) Deduce that﹐if a , b , c and d are positive and p  1 ﹐then


a b


 a 
p 1
(c) Suppose ai i 1 , 2 ,  and
a b
c 

 a 
p
bi i 1 , 2 , 

p
By considering a    a i 
 i 1

n
1
1
p
p 1
d p  c  d  ．
p
are two sequences of positive numbers and p  1 ．
1
p

p 
and b    b j  ﹐
 j 1

n
1
1
p
 n pp  n pp  n
p 
prove that   ai     bi     ai  bi   ﹐
 i 1

 i 1

 i 1

where the equality holds if and only if
a
a1 a 2
a

 n  ．
b1 b2
bb b
page
25
Prepared by K. F. Ngai
Inequalities
AL-93I-1
Advanced Level Pure Mathematics
Prove the following Schwarz’s inequality ﹕
2
 n

 n 2  n 2 
  ai bi     ai   bi  ﹐where a i ， bi  R and n  N ．
 i 1

 i 1
 i 1 
Hence﹐or otherwise﹐prove that
AL-01I-3
1 n
1 n 2
ai 

 ai ．
n i 1
n i 1
(a) Let 0    1 . Show that x   (1   )  x for all x  0 .
1 1
(b) Let a, b, p and q be positive numbers with
  1.
p q
1
p
1
q
Prove that a b 
a b
 .
p q
page
26
Prepared by K. F. Ngai