[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.

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1 [ 0 ]. Iequlity eists oly betwee two rel umbers (ot comple umbers).. If be y rel umber the oe d oly oe of there hold.. If, b 0 the b 0, b 0.. (i) b if b 0 (ii) (iii) (iv) b if b b if either b or b b if either b or b 0, 0, 0 Iequlities 5. I give iequlity, terms/coefficiets from oe side to other side c be trsferred s i the cse of equlity. 6. O c dd/subtrct the sme rel umber o both sides of iequlity, the directio of iequlity does ot chge. 7. Two is equlities with sme directio c be dded (lwys) d multiplied (if both sides of the iequlity re positive). But they c ever be subtrcted or divided. 8. Both sides of iequlity c be multiplied by sme positive qutity without chgig the directio of iequlity. 9. The directio of iequlity chges if it is multiplied both sides by egtive umber. 0. If b the c bc if, b, c 0.. If b the f ( ) f ( b), if f() is icresig fuctio of d lso if b, the f ( ) f ( b) if f ( ) is decresig fuctio of.. For closed cove polygo i XY-ple, y lier fuctio of d y i.e., z by defied over such cove polygo will hve mimum d miimum vlue oly t the vertices of the polygo.. If c d b c the b c or b. Also if b d b c the c.. Fuctios f (, b, c), g(, b, c ) d h(, b, c ) c ot be positive simulteously if f g h 0 0

2 [ ] y f ( ) f ( y) 5. Jese s Iequlity : If f for the f ( ) is positive d curves towrds -is t lest i the give domi. y f ( ) f ( y) y If f for y the the fuctio curves wy from -is.,, [, b] ( y), y, [, b] 6. Method of iductio is very useful i provig result of iequlity ivolvig oly turl umber. 7. If,,..., the. 8. If b, b,..., b the... b b... b d equlity sig holds iff b, b,..., b. 9. (i) If b 0 the. b (ii) If b 0 d c d 0 the b d c bd. d c b (iii) If b 0 d d 0 the. The equlity sig holds iff b d c d. d d 0. (i) Let b 0, p, q 0 d let the /q d p/ q p/ q b d b /q b deote positive q th roots of d b respectively p/ q p/ q (ii) Let b 0, p, z e o egtive iteger d q positive iteger d roots of d b respectively the p/ q p/ q b d b The equlity sig holds of d oly if p/ q p/ q b or p 0. / q / q, b deote q th. (i) For two positive umbers d b, let rithmetic deotes A. me, G deotes geometric me d H deotes Hlmout me the A G H. (ii) The equlity sig holds if d oly if b. If A, G d H be respectively the rithmetic me, the geometric me d Hrmoic me of positive itegers,,..., the A G H. The equlity sig holds if d oly if..... (i) If is rel d A B C 0 (ii) If A 0 d is rel the B A B C 0 AC B 0. Trigle Iequlity : b b b... i i i i AC 0.

3 [ ] 5. The legths, b, c c represet the sides of trigle if d oly if b c, b c, c b. 6. Weierstrs s Iequlity : For positive umber,,,..., If i re frctios (less th oe) the 7. Cuchy Schwrtz Iequlity : ( )( )( )... ( ) c... ( )( )... ( ) (... ) ( b b... b ) (... )( b b... b ). 8. Tchebychev Iequlity : If... d y y y... y or... d y y y... y. the y y... y... y y... y If oe of the sequeces is icresig d other is decresig the the directio of the iequlity chges. 9. If {,,..., } re positive umbers, b { b i } re vrious permuttios of i the i ibi i i i.e., 0. The product ttis mimum. Also usig AM. k l m d whe b c d Z will be mimum whe GM klm k l m z k l m d k l m k l m. k l m b c d k l m pq p p p q q q q p. Holder s Iequlity : ( b b... b ) (... ) ( b b... b ) Where, i d b i re o egtive rel umbers. P q Emple. Solutio : For y three positive rel umbers, b, d c, show tht We kow tht by squrig we hve b b b b b b b b b c b bc c.

4 [ ] Emple. Similrly b b b Addig these three reltios, we get Show tht b b c, c bc c b c b c b bc c ( b c ) b bc c. Hece proved. y z y z y z. Solutio: Let the three umbers be, y d z. Applyig A.M.,G.M. iequlity to d. y y y y y () Similrly, Ad Addig (), () d (), we get z z () y z y z () y z y z y z. Hece proved. Emple. Solutio : If y z 7 show tht y z 8. Applyig Cuchy-Schwrtz iequlity, to sets / / / (, y, z ) d / / / (, y, z ), We hve ( y z ) ( y z )( y z) () Agi pplyig C-S iequlity, to (, y, z ) d (,,). We hve Squrig both sides of () we hve ( y z) ( y z ) () ( y z ) ( y z ) ( y z) O usig (), the bove iequlity yields, ( y z ) ( y z ) ( y z ) ()

5 [ ] Sice y z 7, from (), we get Tkig +ve squre root, ( y z ) (8) y z 8 Hece Proved. Emple. Show tht ( p qy)( pq y) pqy; give p, q,, y re +ve. Solutio : Give p, q,, y re +ve p, qy, pq, y re +ve umbers Applyig A.M., G.M. iequlity to p d qy. p qy pqy p qy pqy () Agi, Applyig A.M. G.M. iequlity to pq d y pq y pqy pq y pqy () Multiplyig () d () we get ( p qy)( pq y) pqy. Emple 5. Solutio : Emple 6. Solutio : Show tht If b c bc., b d c we get Applyig A.M. G.M. iequlity. b c If, b, c re uequl d positive show tht, b, c re uequl d positive. Applyig A.M. G.M. iequlity to Similrly, b c bc. Hece Proved b d b c b bc c.. b b b b () b c bc () c c ()

6 [ 5 ] Emple 7. Add to get If be positive umber, prove tht b c b bc c. Solutio : Let two positive umbers be d. By A.M., G.M. Iequlity Emple 8. Prove tht : or or.. H.P. Solutio : Let the positive umbers be By A.M. G.M. Iequlity, Emple 9. Proved tht t cot. Solutio : d. Applyig A.M. G.M. iequlity to. Hece Proved t t t t t t t cot (By A.M. G.M.) 5

7 [ 6 ] Emple 0. Which is bigger Solutio : Now 7 or ( ) Thus Emple. Sow tht Solutio : ( ) () is divisible by ( ) ( ) () ( )(..... ) 5 p Similrly, q d bove q thus the epressio is divisible () is divisible by 5 Emple. For positive, b, c prove tht. Solutio : b b c c bc( b c) Let the two sets of umbers be By Tchebycheff iequlity A {, b, c} B { b, b c, c } ( b) b( b c) c( c ) ( b c) b b c c by A.G. G.M. Iequlity ( b b c c ) ( b c)( b b c c ) () b b c c Hece from () d (), we get b b c c bc () b b c c ( b c)( bc). 6

8 Emple. For positive, b, c prove tht Solutio : As, b, c re positive. [ 7 ] b c c b We my ssume tht b b c c Hece b b c c Let A { b, b c, c } Now pply Thebycheff s iequlity B,,. b b c c b b c c {( b) ( b c) ( c )} b b c c ( b) ( b c) ( c ) ( b c) b b c c b c b c b c b b c c 9 c b 9 b b c c b c c b bc c b Emple. For positive, b, c such tht bc show tht. Solutio : From the give epressio b c c b c bc ( ) ( b) ( c) c c b b c ( c ) ( c ) b c c c c bcc cb b cb We c ssume tht b c s the equtio is symmetric. So the umbers re positive itegers rised to positive powers. So the deomitor is greter th the umertor. Hece the bove epressio is. b c c b i.e., b c. [ bc ] 7

9 [ 8 ] Emple 5. Detemie ll rel umber stisfyig the iequlity, log( ) log ( 9). Solutio : log( ) log ( 9) log ( ) log 9 / log[( )( 9)] log0 / [( )( 9)] (0) But for log to be defied d 9 0 d > 9. i.e., 9. Hece the required vlue 9 As. Emple 6. If, b, c, d re positive rel umbers the show tht Solutio : Without loss of geerlity, b c d. d Applyig Tchebycheff s iequlity or Emple 7. If, b, c, d re +ve, prove tht d b c d d d ( b c d) d ( b c d) 6. d d bcd( d). ( b c d) 6. d 8

10 [ 9 ] Solutio : We choose two sets Applyig iequlity (, b, c, d )d (, b, c, d ) ( b c d ) ( b c d )( b c d ) Applyig A.M. G.M. iequlity to b c d, b, c, d d b c d bcd d bcd ( d). Emple 8. Show tht ( ) Solutio : Cosider the umbers A.M. G.M [ ] 6... ( ) 6... ( ) Emple 9. For positive umbers, y, z Show tht Solutio : Cosider the umbers : ( y z) 7( y z )( z y)( y z) y z, z y d y z y z ( y z )( z y)( y z) ( y z) 7 ( y z )( z y)( y z) ( y z) 7( y z )( z y)( y z) Emple 0. Show tht... if,,, re positive. Solutio : Cosider the umbers:,,...,, 9

11 [ 0 ] A.M. A.M....,,,...,..... Emple. Show tht.,..., ( ). Solutio : Cosider the umbers,, 5,..., ( ) ( ) ( ) [ ] ( ) ( ),, 5,..., ( ) Emple. Fid the miimum vlue of y subject to the coditio y 8, d y. Solutio : Applyig A.M. G.M. iequlity y y y y 8 Miimum vlue of y 8.. y Emple. Let, b, c, d be rel umbers such tht b c d..8 R Solutio : Prove tht ( b c d ) 8( c bd) Cosider the qudrtic polyomil with rel coefficiets f ( ) ( )( c) ( b)( d) or f ( ) ( b c d ) ( c bd) sice b < c d, f ( ) 0, f ( b) 0, f ( c) 0, f ( d) 0 Hece f ( ) 0 hs rel root betwee d b d lso betwee c d d (by Descrte s rule of sigs) i.e., the qudrtic equtio f ( ) 0 hs roots which re rel d distict. Discrimit is positive i.e., ( b c d) ( c bd) i.e., ( b c d ) 8( c cd ) 0

12 Emple. Without usig tbles, prove tht Solutio : Let log i.e., d log5 [ ] log log b 5 d d 5 b / d 5 5 b 0 b But 0. (Sice ) 7 or Show tht log log b b b Emple 5. If, b, c re three umbers 0, Solutio : such tht b c, prove tht Now b bc c b b b c bc c c / b b c b bc c () ( b c) ( b bc c) b bc c i.e., ( b bc c) b bc c. Emple 6. Let, b, c be rel umbers with 0, 0 b, 0 c d b c. Prove tht 8. b c

13 [ ] Solutio : If, y, z re positive, the y y, y z yz z z ( y)( y z)( z ) 8yz () Thus 8( )( b)( c) [( b)( b c)( c )] 8( )( b)( c) ( b)( b c)( c ) cb.. 8 b c Emple 7. Show tht, for trigle with rdii of circum circle d icircle equl to R d r respectively, the iequlity R r hols. Solutio : bc R d r s R bcs bcs r. s( s )( s b)( s c) bc ( b c )( c b)( b c) Now, pplyig A.M. G.M. iequlity. ( b c ) ( c b) ( b c )( c b) () i.e., c ( b c )( c b) () similrly, b ( c b )( b c) () ( c b)( b c) () bc ( b c )( c b)( b c) (5) bc ( b c )( c b)( b c) bc ( b c )( c b)( c b c) (6) (7) R i.e., r or R r (8) Emple 8. Prove tht Solutio : Let s For y, d hece (00 ) 00

14 [ ] s times Agi, rte, terms hudred t time Hece S s Emple 9. If bcd show tht ( )( b )( c )( d ) 6. We epd ( ) b )( c )( d ) d collect the terms i pirs such tht the product of the terms i ech pir is. Thus, ( )( b)( c)( d) ( bcd) ( bcd) ( b cd) where the iteger writte uder the (symbol) deotes the umber of terms govered by Now, ech of the terms is of the firm ( bcd ), ( bcd) d ( b cd) (with 0) d so it is. Sice there re 8 such terms, we hve the sum ( bcd ) ( bcd) ( b cd ) ( c bd ) ( d bc) ( b cd ) ( b cd ) ( c bd) ( bc d) 8 y z

15 [ ] Emple 0. Show tht 0 the Solutio : Sice, 6 0. pplyig A.M. G.M. iequlity, Thus Emple. Show tht if, y, z re o-egtive rels, such tht y z. Solutio : y z, the 8 y z lso y z yz ( y z) y z ( y z)( z )( y) 8 yz z y 8yz y z z y.. 8 y z Emple. If, b, c, d re positive rel umbers. Show tht Solutio : ( b c d) 6. d b c d / / ( bcd ) d d.. d Multiplyig LHS d RHS respectively. ( b c d) 6. d Emple. If, b, c re positive rel umbers, prove tht, b c c b ( b c) b c b Solutio : Now b c ( b c ) (i) b c b c Similrly b c b c c c ( b c ) (ii) b ( c ) d b ( b ) (iii)

16 [ 5 ] Addig b c c b ( b c ) b c c b ( b c). Emple. Show tht (!). Solutio : Cosider the uequl positive umbers,,,...,..e., i.e., Risig both sides to powers,... ( ) ( ) (!) / (..... ) (A.M. G.M.) / [(!) ] i.e., (!) Emple 5. If, b, c re positive d b c. Show tht 9. Solutio : O dividig by ( b c) successively, by, b, c we get Addig, Emple 6. Show tht log0. b c (i) c (ii) b b b b (iii) c c c b b c c b c b c 9 Solutio : 9 0 log0 9 log0 0 0 log () i.e., log0 0 log (i) 5

17 [ 6 ] 7 0 log0 7 log0 0 i.e., 0 log () 0 0 Thus log0. Emple 7. I ABC, Show tht. b c c b Solutio : Now Multiplyig L.H.S. i.e., log log (ii) ( b) ( b c) ( c ) / {( b)( b c)( c )} (i) b b c c b b c c 9 ( b c) b b c c 9 b c c b Also b c i.e., Thus Equlity occurs whe b c. b c b c b c b c / ( b c) b c c b b c b c c b (ii) (iii) (iv) b c c b Emple 8. If, b, c re sides of trigle show tht. b c Solutio : Sice, b, c re the sides of trigle, b c 0, b c 0, c b 0 (i) b c c b Thus,, re ll positive (ii) b c b c Tke,... times 6

18 [ 7 ] c c,... b times b b b b,... times c c d pply A.M. G.M. iequlity b c c b b c b c LHS of the iequlity cbc b c c b b c i.e., R.H.S. b c c b Thus. b c Emple 9. If, b, c, d re four o-egtive rel umbers d b c d, show ht Solutio : b bc cd. ( b c d) ( b bc cd ) b c d b bc cd c d bd b b c d cd bc c d bd ( b) ( c d) ( b)( c d) d Aliter : [( b) ( c d)] d 0 (, b, c, d 0) ( b bc cd ) 0 ( b bc cd ) ( b bc cd ) the bove problem c be solved by usig A.M. G.M. iequlity we kow tht ( c) ( b d). ( c)( b d) ( c) ( b d) ( c)( b d) ( c)( b d) b d bc cd 7

19 [ 8 ] b bc cd d. Emple 0. If, b, c, d re positive, the prove tht b bc cd. ( d 0) Solutio : ( b b c c d d )( b bc cd d ) 6( bcd ). Applyig Cuchy Schwrz iequlity,,, b c d d d we get b b, bc b, cd b, d b, b b, b b c,, b d. Now pplyig A.M. G.M. iequlity, we get ( b b c c d d ) d 6 d d hece Emple. Give tht ( b b c c d d )( b bc cd d ) 6( bcd ). y z 8, prove tht Solutio : y z 6. Applyig Cuchy Schwrtz iequlity with Let / / /, y, z d / / /, y, z we hve ( y z ) ( y z )( y z) Agi y z y z so ( y z) ( y z )( ) ( y z) 8 d hece ( y z ) 6 ( y z ) ( y z). 8 6 y z. Emple. If y z 6. w y z 0, show tht w y z 500 8

20 [ 9 ] Solutio : Applyig Cuchy Schwrz iequlity for w,, y, z d w,, y, z, we get ( w y z ) ( w y z )( w y z ) () Agi pplyig Cuchy Schwrz iequlity with ( w y z ) ( w y z ) w,, y, z d,,,, we get ( w y z ) ( w y z ) () ( w y z ) ( w y z ), ( w y z ) by Eq. () / 00 ( w y z ) 50 / ( w y z ) ( w y z ) / w y z 50 or 500, by () Emple. If ( ) (5 ) d ( ) (7 ), fid the itegrl vlues of. Solutio : We hve ( ) 5 ( ) (5 ) 0 0 ( )( ) 0 () d Agi ( ) (7 ) 5 0 ( )( ) 0 () () Agi Eq. () d Eq. () cot stisfy oe other. Hece, we should cosider Eq. () d Eq. () from which we get Emple. Prove tht the polyomil Solutio : Let d so.... is divisible by P d Q

21 [ 0 ] P Q ( ) ( ) ( )... ( ) [( ) ] [( ) ] [( ) ]... [( ) ] () But 0 ( ) is divisible by 0 for ll RHS of Eq. () is divisible by P Q is divisible by 0 0. d hece divisible by Emple 5. Give tht the equtio pr 6s 0, (ii) q 6s 0. p q r s 0 hs four positive roots, prove tht (i) Solutio : Let,,, be he four positive roots of the give equtio. The, (i) (ii) p () g q () r () s () Usig A.M. G.M. iequlity i Eq. () d Eq. (), we get. r. p s s pr 6s or pr 6s 0 Applyig A.M. G.M. iequlity i Eq. (), we get q q 6 6 6s or q s 6s 0. Emple 6. If, b d c re positive rel umbers such tht b c, prove tht ( )( b)( c) 8( )( b)( c). Solutio : We kow b c b c d ( b c) ( b) ( c) d sice b c where, b d c re positive rel umbers, so b d c re positive. Applyig A.M. G.M. iequlity, we get ( b) ( c) ( b)( c) () 0

22 [ ] b ( ) ( c) ( )( c) () d c ( b)( ) ( b)( ) () Multiplyig Eq. (), Eq. () d Eq. (), we get ( )( b)( c) 8( )( b)( c). Emple 7. If, b d c re the sides of trigle d b c, the prove tht b c bc. Solutio : We kow b c d squrig, we get ( b c) b c ( b bc c) b c ( b bc c) Addig bc if both sides, we get b c bc ( b bc c bc) To prove b c bc, it is eough to prove tht ( b bc c bc) or bc b bc c) or b bc c bc 0 b c s s Emple 8. For N,, show tht Solutio : We hve ( ) terms ( ) terms ( ).... Emple 9. Let, b, c be rel umbers with 0, b, c d b c. Prove tht Solutio : Here we use A.M. G.M b c ( b c) ( b c) ( b c)( b c)

23 [ ] ( b c) ( b c) b ( b c)( b c) ( c b) ( c b) c ( c b)( c b) [( b c) ( b c)][( b c) ( b c)][( c b) ( c b)]. b. c. 8 ( b c)( b c)( b c)( b c)( c b)( c b) ( c)( )( b) 8( c)( )( b). [ b c ]. 8. b c

SM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory

SM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory SMH Uit Polyomils, Epoets, Rdicls & Comple Numbers Notes.1 Number Theory .1 Addig, Subtrctig, d Multiplyig Polyomils Notes Moomil: A epressio tht is umber, vrible, or umbers d vribles multiplied together.