RI Chapter 7 Inequalities

RAFFLES INSTITUTION H3 Mathematics 9820 _____________________________________________________________________________________________ _________________________ Chapter 7: Inequalities Page 1 of 18 Chapter 7 Inequalities The study of inequalities is important in mathematics. For starters, we often need to compare the magnitude of two mathematical objects. This is essentially with the aid of inequalities. You will find that in university, the use of inequalities in analysis is particularly important. For example, in describing the concept of a “limit” and “convergence”. SYLLABUS INCLUDES  Equations and inequalities (such as Triangle inequality, AM-GM inequality, Cauchy- Schwarz Inequality) CONTENT 1 Introduction 1.1 Estimation 1.2 Absolute values, estimating size 2 Triangle Inequality 3 Arithmetic-Mean and Geometric Mean (AM-GM) inequality 4 Cauchy-Schwarz Inequality 5 Inequalities involving other functions 5.1 Inequalities involving monotonic functions 5.2 Inequalities involving convex functions

Raffles Institution H3 Mathematics _______________________________________________________________________________________________________ _______________________ Chapter 7: Inequalities Page 2 of 18 1 Introduction In general, inequalities are used to make comparisons, and to make estimations. It is one of two simple tools that form the language of estimations, the other being absolute values for measuring size and distance. 1.1 Estimations One of the major uses of inequalities, as we mentioned, is to make estimations. These have a language of their own that you need to get used to. Definition 1.1.1 If c is a number we are estimating, and m < c < M, we say that m is a lower estimate (lower bound) for c and M is an upper estimate (upper bound) for c. If two sets of upper and lower bounds satisfy the inequalities ''m m c M M    we say that ', 'mM are stronger or sharper estimates for c, while m, M are weaker estimates. Estimates are obtained in many ways; the inequality laws that you have been introduced to in H2 Mathematics usually play an important role. Calculus can also be used. Here are some examples to illustrate. Example 1 Give upper and lower bounds for 42 1 31aa . Solution We are not told what a is, so our bounds will have to be valid no matter what a is, i.e. valid for all a. Let us start with the denominator. Since any square is nonnegative, the polynomial has its smallest value 1 when a = 0, and it has no upper bound since a can be arbitrarily large. Hence we have 421 3 1aa   which implies 42 101 31aa  These are the sharpest possible estimates valid for all a: In fact, the upper bound 1 is attained when a = 0, and the lower bound is 0 since the fraction can be made arbitrarily close to 0 by taking a sufficiently large. Exercise 2 Give upper and lower estimates for 2 2 1 sin 1 cos n n   , for 0n .