F. Inequalities. HKAL Pure Mathematics. 進佳數學團隊 Dr. Herbert Lam 林康榮博士. [Solution] Example Basic properties

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1 進佳數學團隊 Dr. Herbert Lam 林康榮博士 HKAL Pure Mathematcs F. Ieualtes. Basc propertes Theorem Let a, b, c be real umbers. () If a b ad b c, the a c. () If a b ad c 0, the ac bc, but f a b ad c 0, the ac bc. Theorem Let a ad b be postve umbers. The a b f ad oly f a b. The proof of above s smple ad s left to the studet. Example (a) Prove the eualty, where 0. (b) Hece show that (c) Show that s close to f s suffcetly large. S [Soluto] (a) ( )( ). 85

2 ,.e.,. Hece. Smlarly, we ca show., (b) For, we have. But ths eualty ca be modfed as (*) 86 For,,,00, we have Addg up (*) ad the others correspodg to,,,00, we obta As , therefore we have e

3 (c) Smlar to what we dd (b), cosder the eualtes. Addg up these, we have ( ) S ( )., We obta S ( ) ( ). Hece, S S S S. Ths eualty mples that S s close to large. For example, whe,000,000, less tha. as becomes S devates from Well kow eualtes The tragle eualty, Cauchy-Schwarz s eualty ad the eualty cocerg arthmetc mea ad geometrc mea are preseted the followg examples. Example Show that ab a b. 87

4 [Soluto] Startg wth ab ab, we have a abb a ab b. Sce a a, b b, ab a b, the above eualty ca be wrtte as a abb a a b b ( a b) ( a b). ab a b. Eualty holds f a b. Note: Ths s kow as the tragle eualty, ad s also vald whe a ad b are complex umbers or vectors. Ths wll also be dscussed Chapter 4. The geometrc terpretatos for these two cases are the same, stated as follows. The sum of ay two sdes of a tragle s loger tha the thrd oe, e.g. PQ QR PR Fgure. Fgure Example Show that where 88 ( ab ab ab ) ( a a a ) ( b b b ), a 's ad b 's are real umbers. Ths s kow as the Cauchy-Schwarz s eualty.

5 [Soluto] Cosder ax b,,,,. We have Let The ( ax b) 0, for all x. ( a x abx b ) 0 a x ab x b a B ab, C b. A, 0. Ax Bx C 0. The reured eualty s trval whe A 0. Suppose A 0. For the uadratc expresso to be o-egatve ( B) 4AC 0.e. B AC ab a b. Geometrcally, the graph of Ax Bx C s above the x -axs. The eualty holds ff ( ax b) 0 for some ozero x. a a a Thus ax b 0,,,,,.e.. b b b x Example 4 Gve postve umbers a, a,, a, show that the arthmetc mea (A.M.) s greater tha or eual to the geometrc mea (G.M.),.e. a a a aa a. [Soluto] Proof of ths well kow eualty reures usg the property of the atural logarthmc fucto, l( xxx) l xl x l x, where x 's are postve. 89

6 Frst, we have to show whe x 0, l x x (*) From Fgure, we observe that the curve of y l x s below the le y x, ad they touch each other at x. Fgure O the other had, we ca cosder the fucto f ( x) lxx. Sce f '( x) 0 x, x ad f ''( x) f ''( x) 0, x the curve of f ( x ) attas a local maxmum at x ad the maxmum value s zero. Hece, whe x 0, f( x) 0,.e. l x x. aa a Secod, let A, a 's 0. a a By (*), we have l, A A a a l, A A a a l. A A Addg up above, a a a a a a A A A A A A l l l aa a a a a. A A l Note that RHS of the eualty 0, aa a aa.e. l 0 a A A 90

7 aa a aa a aa a aa Eualty holds f a a a. a. A alteratve method of provg ths eualty s to use the followg eualty (**). If xx x, the x x x. (**) a Let x aa a ad we have x. The a a a x aa a aa a, aa a. Note: () (**) ca be proved by ducto o. (See Example 4 of Secto A, Chapter, o P.07 ) Otherwse, we ca smply duplcate the above proof usg the atural log fucto. l x x, l x ( x ) Sce xx x, LHS 0. Hece x. l x x. () A.M. G.M. s freuetly used to prove eualtes, ad s cosdered more effcet tha usg algebrac mapulatos certa occasos. (see Example 6 o P.9) 9

8 Example 5 (a) Show that the sum of a postve umber ad ts recprocal s always greater tha or eual to,.e. x for x 0. x (b) Let p ad be postve umbers. Show that. p p 4p [Soluto] (a) It mmedately follows from A.M. G.M. as x x x x. x x Alteratvely, x x 0 x, x x. x (b) It s euvalet to show By (a), p p 4. p p p ad p p p p 4,.e. p p p p 4. p Example 6 Let a, b, c be dstct postve umbers. (a) Show that a b a b ab. (b) Show that abbabccbcaac 6abc. (c) Hece, show that a b c abc. [Soluto] (a) ( a b 0) ( a b)( a b) 0 a b a bab 0, a b a b ab. 9

9 (b) ( a b) 0, a b ab,. a c b c abc Smlarly b a c a abc, c b a b abc.. ab ba bc cb ca ac 6abc (c) From (a) Smlarly, a b a b ab. b c b c bc, c a c a ca. The ( a b c ) a bab b cbc c a ca. From (b) abbabccbcaac 6abc. Ths mples ( a b c ) 6abc, a b c abc. Note: Ths eualty ca be proved a dfferet way. Sce a, b ad c are dstct postve umbers, so are a b c The mples a b c abc. a, abc (A.M.>G.M.) b ad c. Example 7 Let a, b ad c be postve umbers. Usg A.M. G.M., show that ( )( )( ) ( ) a b c abc. Uder what codto o a, b ad c wll the eualty hold? [Soluto] Usg A.M. G.M., ( ) abc abc ad ( ) ( )( )( ) ( ) ab bc ca ab bc ca abc. 9

10 Hece, ( a)( b)( c) ( abc) [ ( ) ( ) ] ( ) ( a b c) abc ( ab bc ca) ( ab)( bc)( ca) 0 a b c ab bc ca abc abc abc abc Eualty holds f ad oly f ( ) abc abc ad ( ab bc ca ) ( ab )( bc )( ca ) a b c ad ab bc ca,.e. a b c.. Ieualtes volvg polyomals (a) Roots of a polyomal Suppose we are able to fd the roots of the polyomal euato f( x) 0 (*). The f( x) 0 & f( x) 0 ca be solved mmedately. Cosder the graph of f ( x ) show Fgure. It ca be see whe x x x or x x, the f( x) 0, ad whe x x or x x x, the f( x) 0. Fgure (b) Extrema of a polyomal By fdg the maxmum ad mmum pot of a polyomal, we are able to show f ( x) k or f ( x) k for x some specfed tervals. Ths ca be vsualzed by cosderg the local mmum P ( a, b ) ad local maxmum Q (, c d ) Fgure. 94

11 Example 8 Solve x x. [Soluto] Suarg both sdes, Solvg, ( x ) ( x ). ( x ) ( x) 0 [( x ) ( x)][( x ) ( x)] 0 ( x x )( x x 4) 0 (*) The roots of the buadratc euato ( x x)( x x4) 0 are foud by solvg two uadratc euatos, x x 0, x x4 0 x,, Therefore, the graph of follows 7 x. f( x) ( x x)( x x 4) s sketched as Fgure 4 Observe that eualty (*) s satsfed 7 7 whe x or x. Example 9 Show that 4 x x x x whe x. 95

12 [Soluto] 4 Let y f( x) x 8x x 4x 8. Dfferetatg f ( x ), f '( x) 4x 4x 44x 4 4( x 6x x 6). By factorzato, x 6x x6 ( x)( x 5x 6) ( x)( x)( x ). Puttg f '( x) 0, we obta the turg pots x,,. Correspodgly, y, 0,. f ''( x) 4(x x ) The f ''() 4( ) 0 local mmum at x. f ''() 4(4 4 ) 0 local maxmum at x. f ''() 4(9 ) 0 local mmum at x. Hece, we ca see the graph of f ( x ) s below the x -axs [, ],.e. 4 x 8x x 4x8 0, whe x. Fgure 5 Example 0 Let x ad 0. (a) Prove that ( x) x. (b) Hece, show that for postve umbers p &, 96

13 p p ( ). [Soluto] p (a) Suppose s ratoal. The 0 dcates where p & are postve tegers ad p. Thus, ( x) ( x) ( x) p p ( x )( x ) ( x ) p terms ( p) terms ( x) ( x) ( x) p( x) ( p) px ( G.M. A.M.) p x x. By the way, the eualty holds whe x 0. For the case of beg rratoal, we use a seuece of ratoal umbers { r } the terval (0,), whch coverges to,.e. lm r. r Sce ( x) rx, 0, we obta r ( x) lm( x) lm( r x) x. r Obvously, eualty holds whe x 0. A alteratve method s to use dfferetal calculus. Let f ( x) ( x) x. Dfferetatg ad puttg the frst dervatve eual to zero, f '( x) ( x) 0 x 0. Moreover, f ''( x) ( )( x), f ''(0) ( ) 0 ( 0 ) f ( x ) has a local maxmum at x 0. Hece f ( x) f(0) x 97

14 (b) Note that ( x) x 0, [ f (0) 0] ( x) x. p ca be expressed as x where x. p The x p x. By (a), p p. Multplyg both sdes by, p p. p p ( ) 98