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Inequality - Inequalities Advanced Level Pure Mathematics...

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Inequalities Advanced Level Pure Mathematics Advanced Level Pure Mathematics 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 Fundamental Concepts of Inequalities and Methods of Proving Inequalities Algebraic Inequalities 1. For any real number a , . 0 2 a Equality holds iff 0 = a . 2. If b a and , c b then . c a 3. If b a , then . , R c c b c a 2200 + + page 1 Prepared by K. F. Ngai 6
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Inequalities Advanced Level Pure Mathematics 4. If b a , then 0 , 2200 c bc ac and 0 , < 2200 < c bc ac . 5. If , 0 b a then p p b a for any . 0 p Equality holds iff . 0 = q Example 1 (a) Prove that for any , 0 , y x . 2 ) ( 2 2 2 xy y x y x + + (b) Hence or otherwise, deduce that if , 0 , , c b a then (i) ) ( 2 2 2 2 2 2 2 c b a a c c b b a + + + + + + + (ii) abc a c c b b a 8 ) )( )( ( + + + Solution (a) (i) 0 ) ( 2 2 - = - + y x xy y x xy y x 2 + (ii) 2 2 2 ) ( ) ( 2 y x y x + - + = 2 2 2 2 2 2 2 y xy x y x - - - + = 0 ) ( 2 - y x 2 2 2 ) ( ) ( 2 y x y x + + y x ) y (x + + 2 2 2 ) 0 , ( y x (b) (i) By (a), we have ) ( 2 1 2 2 b a b a + + , ) ( 2 1 2 2 c b c b + + and a) (c a c + + 2 1 2 2 Adding up the inequalities, we have 2 2 2 2 2 2 a c c b b a + + + + + ) ( 2 1 a c c b b a + + + + + ) ( 2 ) ( 2 2 c b a c b a + + = + + = (ii) By (a) we have ab b a 2 + ) 0 ( , bc c b 2 + ) 0 ( ca a c 2 + ) 0 ( Multiplying the inequalities, we have ) )( )( ( a c c b b a + + + ca bc ab 8 abc 8 = Example 2 Given that . , , R c b a Prove that ca bc ab c b a + + + + 2 2 2 Example 3 Prove that , 0 , 2200 y x y x y x + + 4 1 1 . page 2 Prepared by K. F. Ngai
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Inequalities Advanced Level Pure Mathematics Equality sign holds iff . y x = Example 4 Show that , , R b a 2200 (a) 2 2 2 ) ( 2 1 b a b a + + (b) 4 1 ) 1 ( - a a Example 5 Show that ) , 0 ( , sin π 2200 < x x x . Example 6 Show that R y x y xy x 2200 + + , , 0 3 5 3 2 2 . Example 6.9 If c b a , , denote the lengths of the sides of a triangle, prove that 4 ) )( ( < - + + + bc a c b c b a Solution page 3 Prepared by K. F. Ngai
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Inequalities Advanced Level Pure Mathematics Example 6.10 Let n m a , , be positive real numbers not equal to 1 and n m . Prove that n n m m a a a a - - + + Solution Example 6.11 Let b a , and c be real numbers such that 1 = + + c b a .Prove that 3 1 2 2 2 + + c b a When does equality hold? Solution Example 6.12 Let b a , be positive real number. Prove that a b b a b a b a Solution page 4 Prepared by K. F. Ngai
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Inequalities Advanced Level Pure Mathematics Example 6.18 Show that for any positive integer k , k k k 2 1 1 < - + Hence deduce that for any positive integer n , ) 1 1 ( 2 1 3 1 2 1 1 - + + + + + n n Solution Example 6.14 Let n be a positive integer. Prove that 1 2 1 2 2 1 2 + - < - n n n n Solution page 5 Prepared by K. F. Ngai
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Inequalities Advanced Level Pure Mathematics Example 6.19 Show that for any positive integer k , 1 3 2 3 2 1 2 + - - k k k k Hence deduce that for any positive integer n , 1 3 1 ) 2 1 2 ( ) 6 5 )( 4 3 )( 2 1 ( + - n n n Solution 6.4 Arithmetic Mean and Geometric Mean Theorem 6-3 G.M. A.M. Let x and y be two positive real numbers. We have xy y x + 2 and equality holds if and only if y x = Proof page 6 Prepared by K. F. Ngai
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Inequalities Advanced Level Pure Mathematics Equality holds if and only if 0 = - y x i.e. y x = y x = Definition Let n
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