HKAL PURE MATHEMATICS
Inequalities
Advanced Level Pure Mathematics
Advanced Level Pure Mathematics
Algebra
Chapter 6
InequalitiesFundamental 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
Fundamental Concepts of Inequalities and
Methods of Proving Inequalities Algebraic Inequalities
1.For any real number , Equality holds iff .
2.If and then
3.If , then
4.If , then and .
5.If then for any Equality holds iff
Example 1(a)Prove that for any
(b)Hence or otherwise, deduce that if then
(i)
(ii)
Solution
(a)(i)
(ii)
=
=
(b)(i)By (a), we have
, and
Adding up the inequalities, we have
(ii)By (a)we have
,
Multiplying the inequalities, we have
Example 2Given that Prove that
Example 3Prove that
EMBED Equation.3 .
Equality sign holds iff
Example 4Show that
(a)
(b)
Example 5Show that .
Example 6Show that .Example 6.9
If denote the lengths of the sides of a triangle, prove that
Solution
Example 6.10
Let be positive real numbers not equal to 1 and . Prove that
Solution
Example 6.11
Let and be real numbers such that .Prove that
When does equality hold?
SolutionExample 6.12
Let be positive real number. Prove that
SolutionExample 6.18
Show that for any positive integer ,
Hence deduce that for any positive integer ,
Solution
Example 6.14
Let be a positive integer. Prove that
Solution
Example 6.19
Show that for any positive integer ,
Hence deduce that for any positive integer ,
Solution
6.4Arithmetic Mean and Geometric MeanTheorem 6-3
Let and be two positive real numbers. We have
and equality holds if and only if
Proof
Equality holds if and only if
i.e.
DefinitionLet be non-negative real numbers.
A.M.=
and
G.M.=
are respectively the Arithmetic Mean (A.M.) and Geometric Mean (G.M.) of the given numbers.
Theorem
For any n non-negative numbers
EMBED Equation.3
i.e.
EMBED Equation.3
EMBED Equation.3
The equality holds iff
Example 7By considering the A.M. and G.M. of a suitable set of numbers,
prove the following. ().
(a)
,
(b)
,
Solution
Example 8Give that Prove the following.
(a)
(b)
(c)
Example 9Let be distinct positive numbers, show that
.
Solution
AL-01I-14
(a)If are two real numbers such that , show that and the equality holds if and only if or .
(b)Show by induction that if are positive real numbers such that , then and the equality holds if and only if
(c)Let be positive real numbers. Using (b) or otherwise, show that
EMBED Equation.3
EMBED Equation.3 and the equality holds if and only if .
Example 10For all non-negative integer n, prove that
(a)
, for any
(b)
SolutionExample 11(a)Let be positive numbers. Prove that
Equality holds iff
(b)Hence, or otherwise, prove that ,
(i)
(ii)
Solution
Example 12If , show that
SolutionExample 13By using , show that .
Hence deduce that the roots of the quadratic equation are real and distinct.
Solution
AL-92I-7
(a)Prove that where are positive integers and (b)If are positive numbers and
using and (a)or otherwise prove that
.
Example 14Let be non-negative numbers, prove the following
(a)
(b)
Solution
Example 15Let be positive numbers, prove the following:
(a)
(b)
(c)
(d)
(e)
Note:
Example 16Show that if is a positive integer greater that 1.
(a)
(b)
Example 17If are positive numbers, such that , prove that
if(i)
and are positive numbers.
(ii)
and are positive fractions.
Solution
Example 18Let be a positive integer greater than , prove the following:
(a)
(b)
(c)
(d)
(e)
Solution
Example 19If are positive integers. Show that
(a)
(b)
(c)Hence show
6.5
Cauchy-Schwarz InequalityTheorem
(Cauchy-Schwarz Inequality)
If and are two sets of real numbers, then
.
Equality holds if and only if .
Proof
Define the quadratic polynomial by
, which is non-negative for all real values
i.e.
i.e.
Equality holdsif and only if
if and only if
for any
if and only if
,
if and only if
Example 20Let be real numbers, prove that
Solution
Example 21Let are non-zero real numbers, show that .
Solution
Example 22Let be real numbers..
Show that .
Solution
Example 23Let be a positive integer greater than 1. By using the Cauchy-Schwarz inequality,
show that
Solution
Example 24Given that and Prove that
(a)
(b)
(c)
Solution
Apply the Cauchy-Schwarz Inequality:
(a)Consider the numbers We have
Example 25For any positive number
(a)Prove
Equality holds iff .
Hence, or otherwise, prove that
(i)
(ii)
(b)Prove that for any positive numbers , the following inequality holds.
Hence, or otherwise, if and are three positive real numbers and prove that .
Example 26Prove that for any positive numbers ;
When does the equality hold?
6.6
Absolute ValuesDefinitionThe absolute value of a real number , denoted by ,
is defined by
Properties
(1)
(2)
(3)
and
(4)
(5)
Conversely,if then .
(6)
Conversely,if then
Remark:
EMBED Equation.3 unless
Theorem
For any two real numbers and , the following hold:
(a)Triangle inequality. i.e.
(b)
(c)
Theorem
For any two real numbers and, the following hold:
(a)
(b)For any positive integer , .
(c)
Example 27Solve and
Solution
Consider
Consider
Consider
Consider
or
For all Real number except
The Solution:
and
and except .
Example28
AL-90I-6
Solve the inequality
Solution
AL-93I-7
Find all in satisfying the following two conditions:
AL-90I-12
(a)Let and for all
(i) Find the absolute minimum of on the interval
(ii) Deduce that for all
(b)(i)Let and be positive numbers such that
and
By taking and respectivelyprove thatfor
where the equality holds if and only if
(ii)Deduce thatif and are positive and then
(c)Suppose and are two sequences of positive numbers and
By considering and
prove that
where the equality holds if and only if
AL-93I-1
Prove the following Schwarzs inequality
where and
Henceor otherwiseprove that
AL-01I-3
(a)Let . Show that for all .
(b)Let and be positive numbers with .
Prove that .
6
page 2Prepared by K. F. Ngai
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