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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