1 MT S K THUT S DNG BT NG THC CAUCHY V BT NG THC BUNYAKOVSKI A.MTSQUYTCCHUNGKHISDNGBTNGTHC CAUCHY V BT NG THC BUNYAKOVSKI - Quy tc song hnh:a s cc bt ng thc u c tnh i xng nn chng ta c th s dng nhiu bt ng thc trong chng minh mt bi ton nh hng cch gii nhanh hn. -Quytcdubng:Du=trongbtngthccvaitrrtquantrng.Ngipta kim tra tnh ng n ca chng minh, nh hng cho ta cch gii. Chnh v vykhi gii cc bi ton chng minh bt ng thc hoc cc bi ton cc tr ta cn rn luyn cho mnh thi quen tm iu kin ca du bng mc d mt s bi khng yu cu trnh by phn ny. -Quy tc v tnh ng thi ca du bng:Chng ta thng mc sai lm v tnh xy ra ngthicadu=khipdnglintiphocsonghnhnhiubtngthc.Khip dng lin tip hoc song hnh nhiu bt ng thc th cc du = phi cng c tha mn vi cng mt iu kin ca bin. -Quy tc bin:i vi cc bi ton cc tr c iu kin rng buc th cc tr thng t c ti v tr bin. -Quy tc i xng: Cc bt ng thc c tnh i xng th vai tr ca cc bin trong cc bt ng thc l nh nhau do du = thng xy ra ti v tr cc bin bng nhau. Nu bi ton c iu kin i xng th chng ta c th ch ra du =xy ra ti khi cc bin bng nhau v bng mt gi tr c th. B. MT S K THUT S DNG BT NG THC CAUCHY I.BT NG THCCAUCHY Cho n s thc khng m na a a ,..., ,2 1,2 , > e n Z n , ta lun c: nn na a a n a a a ... . . ...2 1 2 1> + + +Du = xy ra khi v ch khi na a a = = = ...2 1 http://boxtailieu.netboxtailieu.net 2 II. MT S K THUT S DNG BT NG THC CAUCHY 1.K thut tch ghp b s 1.1K thut tch ghp c bn Bi 1: Cho 3 s thc dnga, b, c.Chng minh rng:( )( )( ) abc a c c b b a 8 > + + +Gii: p dng bt ng thc Cauchy, ta c: ( )( )( ) abc ac bc ab a c c b b a 8 2 . 2 . 2 = > + + + (pcm) Bi 2: Cho 4 s thc dnga, b, c, d.Chng minh rng:( )( ) d c b a bd ac + + s +Gii: p dng bt ng thc Cauchy, ta c: ( )( )( ) ( ) ( ) ( )1212121. .=|.|

\|+++++=|.|

\|++++|.|

\|+++s+ +++ +=+ + +d cd cb ab ad cdb abd ccb aad cdb abd ccb aad c b abd ac ( )( ) d c b a bd ac + + s + (pcm) Bi 3: Cho 3 s thc dnga, b, c tha >>c bc a . Chng minh rng:( ) ( ) ab c b c c a c s + Gii: p dng bt ng thc Cauchy, ta c: ( ) ( ) ( ) ( )1 1211212121. .=|.|

\| + +|.|

\| + s|.|

\| + +|.|

\| + s+= + bcacacbcbc bacac abcbc bacac abcabc b c c a c ( ) ( ) ab c b c c a c s + (pcm) http://boxtailieu.netboxtailieu.net 3 Bi 4: Cho 3 s thc dnga, b, c.Chng minh rng:( )( )( )331 1 1 1 c b a abc + + + s +Gii: p dng bt ng thc Cauchy, ta c: ( )( )( )( ) ( ) ( ) ( ) ( ) ( )1111111311 1 1 31111111311.1.1 11.11.111 1 113 333=|.|

\|++++++++s|.|

\|++++++|.|

\|+++++s+ + +++ + +s+ + + +ccbbaaccbbaac b accbbaac b ac b aabc ( )( )( )331 1 1 1 c b a abc + + + s + (pcm) Bi 5: Cho 2 s thc dnga, btha >>11ba .Chng minh rng:ab a b b a s + 1 1Gii: p dng bt ng thc Cauchy, ta c: ( )2 211aba ab a a ab a b a = + s = (1) Tng t:21aba b s (2) Cng theo v (1) v (2), ta c: ab a b b a s + 1 1 (pcm) Bi 6: Cho 2 s thc dnga, b.Chng minh rng:( ) ( )4 216 b a b a ab + s Gii: p dng bt ng thc Cauchy, ta c: ( ) ( )( )( ) ( )( )422222 22. 424. 4 4 . 4 16 b ab a b a abb a ab b a ab + =((

+=((

+s = (pcm) Bi 7: Cho 3 s thc dnga, b, c.Chng minh rng:( ) ( ) ( ) ( )3 31 3 1 1 1 abc abc a c c b b a + > + + + + +Gii: Ta c: ( ) ( ) ( ) ( ) ( ) ca bc ab c b a a c c b b a + + + + + = + + + + + 1 1 1http://boxtailieu.netboxtailieu.net 4 p dng bt ng thc Cauchy, ta c: ( )32333abc ca bc ababc c b a> + +> + + ( ) ( ) ( ) ( )3 3 3233 1 3 3 3 abc abc abc abc ca bc ab c b a + = + > + + + + + ( ) ( ) ( ) ( )3 31 3 1 1 1 abc abc a c c b b a + > + + + + + (pcm) Bi 8: Cho 2 s thc dnga, b.Chng minh rng:1 + + > + + b aabbaabGii: p dng bt ng thc Cauchy, ta c: 2 2 2 2 2 2|.|

\|+ +|.|

\|+ +|.|

\|+ = + +abbaab abba ababbaab12.222.222.22 + + = + + > b aabbaab abba ab(pcm) Bi 9: Cho 3 s thc dnga, b, c tha10 = + + c b a . Tm GTLN ca: 5 3 2c b a A =Gii: Ta c: 337500 5 3 2 15.3.215.3.25.3.2105 5 5 5 5 3 3 3 2 2105 3 2 5 3 25 3 2105 3 2105 3 2= s s|.|

\||.|

\||.|

\| s|.|

\||.|

\||.|

\||.|

\||.|

\||.|

\|> + + + + + + + + + = + + =c b ac b a c b ac b a c c c c c b b b a ac b a Du = xy ra === =+ += = = = + += =532110 5 3 2105 3 2cbac b a c b ac b ac b a Vy GTLN ca A l 337500.1.2K thut tch nghch o Bi 1: Chng minh rng:0 , 2 > > + a,babba Gii:V 0 > a,bnn0, 0 > >abba p dng bt ng thc Cauchy ta c: 2 . 2 = > +abbaabba(pcm) http://boxtailieu.netboxtailieu.net 5 Bi 2: Chng minh rng:1 , 311> >+ aaaGii:p dng bt ng thc Cauchy ta c: ( ) 3 1 2 1111 2 111111= + = + > ++ =+aaaaaa(pcm) Bi 3: Chng minh rng:R e >++aaa, 21222 Gii:p dng bt ng thc Cauchy ta c: 2111 211111 11222222222=++ >++ + =++ +=++aaaaaaaa (pcm) Bi 4: Chng minh rng:0 ,219 1342= s+aaa Gii:Vi0 = a, p dng bt ng thc Cauchy ta c: 213 .3121 3311 393119 13222224242= s+=+=+aaaaaaaaa (pcm) Bi 5: Tm gi tr nh nht ca biu thc:( ) 1 , 211222 = ||.|

\|+++ + = aaaa AGii:

( )( )( )( )( )( )( ) ( )2 2 2 2111 2 2 2111 2111 111 1112 21222222222222+ = +++ +++ + =|.|

\|++ + + + =((

+ + ++ + =||.|

\|+ + ++ + =>aaaaaa aaaaaa aa ACauchy Du = xy ra khi v ch khi( )( )22111 2+= +aahay 28 24 = a Vy GTNN ca2 2 2 + = Ahttp://boxtailieu.netboxtailieu.net 6 Bi 6: Tm gi tr nh nht ca biu thc :0 ,22> + = aaa AGii:p dng bt ng thc Cauchy ta c:

33324232132.2. 21.2.2. 32.2. 212 22= = > + + = + =a aa aa aa aaa ADu = xy ra khi v ch khi 222 aa=hay 34 = aVy GTNN ca 3423= ABi 7: Chng minh rng:0 , 3) (1> > >+ b ab a baGii:p dng bt ng thc Cauchy ta c: ( )( )( )( ) ( )31. . 31 13 = >+ + =+b a bb a bb a bb a bb a baBi 8: Chng minh rng: ( )( )0 , 3142> > >+ + b ab b aaGii:p dng bt ng thc Cauchy ta c: ( )( )( )( ) ( )( )( ) ( )( ) ( ) ( )( )( ) ( )3 121211.21.21. . 4 12121121211442= + ++ + >+ ++++++ =+ +b bb ab bb ab bb ab bb ab b aa 1.3K thut ghp i xng Trong k thut ghp i xng ta cn nm mt s thao tc sau: Php cng:( ) ( ) ( ) ( )+ + + + + = + ++++++= + +a c c b b a c b aa c c b b ac b a22 2 2 Php nhn:( )( )( )( ) => =ca bc ab c b ac b a ca bc ab abc2 2 20 , , , http://boxtailieu.netboxtailieu.net 7 Bi 1: Cho ba s thc dnga, b, c.CMR:c b acabbcaabc+ + > + +Gii: Ta c: c b aabccabcabbcabcaabcabccabcabbcabcaabccabbcaabc+ + = + + >|.|

\|+ +|.|

\|+ +|.|

\|+ = + +. . .212121 Bi 2: Cho ba s thc0 = abc .CMR:cabcabaccbba+ + > + +222222 Gii: Ta c: cabcabcabcabbaacaccbcbbabaacaccbcbbaaccbba+ + > + + = + + >||.|

\|+ +||.|

\|+ +||.|

\|+ = + +222222222222222222222222222222. . 212121 Bi 3: Cho ba s thc dnga, b, c tha1 = abc .CMR: 3 + + + >+++++c b acb aba cac b Gii: ( ) ( ) ( )3 322 2 222 2 23+ + + = + + + >+ + + + + = + + =+ + >||.|

\|+ +||.|

\|+ +||.|

\|+ =||.|

\|+ + = + + >+++++c b a c b a c b ac b a c b a c b aabccabcabbcabcaabcabccabcabbcabcaabccabbcaabccabbcaabccb aba cac b Vy3 + + + >+++++c b acb aba cac b http://boxtailieu.netboxtailieu.net 8 Bi 4: Cho 2, , , ,c b ap b CA a BC c AB ABC+ += = = = A.CMR: ( )( )( ) abc c p b p a p81s Gii: Ta c: ( )( )( ) ( )( )( )( )( )( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )abca c p c b p b a pa p c p c p b p b p a pa p c p c p b p b p a p c p b p a p8122.22.222.2.2=+ + + s + + + s = Bi 5: Cho 2, , , ,c b ap b CA a BC c AB ABC+ += = = = A.CMR: |.|

\|+ + >++ c b a c p b p a p1 1 121 1 1 Gii: Ta c: ( )( ) ( )( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )|.|

\|+ + > + + + + + > + + >||.|

\|++||.|

\|++||.|

\|+=++c b aa p c p c p b p b p a pa p c p c p b p b p a pa p c p c p b p b p a p c p b p a p1 1 122121211 1 11 121 1 121 1 121 1 1 1 1.4K thut ghp cp nghch o Trong k thut ghp cp nghch o ta ng dng bt ng thc sau Vi -eN nv0 ,..., ,2 1>nx x xth ( ) 1..1 1...22 12 1nx x xx x xnn>||.|

\|+ + + + + +Chng minh bt ng thc trn :Ta c vi0 ,..., ,2 1>nx x xth ( )22 12 12 12 1...1. ...1..1 1... nx x xn x x x nx x xx x x nnnnnn= >||.|

\|+ + + + + +http://boxtailieu.netboxtailieu.net 9 Vi3 = nv0 , ,3 2 1> x x xth ( )91 1 13 2 13 2 1>||.|

\|+ + + +x x xx x xBi 1: Cho ba s thc dnga, b, c.CMR:6 >+++++cb aba cac b Gii: Ta c: ( ) 6 3 9 31 1 133 1 1 1= > |.|

\|+ + + + =+ +++ +++ +=|.|

\| ++ +|.|

\| ++ +|.|

\| ++ =+++++c b ac b acb a cba c bac b acb aba cac bcb aba cac b Bi 2: Cho ba s thc dnga, b, c.CMR: 23>+++++ b aca cbc ba (Bt ng thc Nesbit) Gii: Ta c: ( )( ) ( ) ( ) | |2332931 1 12131 1 133 1 1 1= >|.|

\|++++++ + + + + =|.|

\|++++++ + =++ ++++ ++++ +=|.|

\|++ +|.|

\|++ +|.|

\|++ =+++++b a a c c bb a a c c bb a a c c bc b ab ab a ca ca c bc bc b ab aca cbc bab aca cbc ba Bi 3: Cho ba s thc dnga, b, c.CMR:22 2 2c b aa cbc bab ac + +>+++++ Gii: ( ) c b aa cbbc baab acca cbc bab ac+ + ||.|

\|++ +||.|

\|++ +||.|

\|++ =+++++2 2 2 2 2 2 ( ) c b aa cbbc baab acc + + |.|

\|++ +|.|

\|++ +|.|

\|++ = 1 1 1http://boxtailieu.netboxtailieu.net 10 ( ) c b aa cb a cbc ba c bab ac b ac + + |.|

\|++ ++|.|

\|++ ++|.|

\|++ += ( ) ( )c b aa cbc bab acc b a + + |.|

\|++++++ + =( ) 1|.|

\|++++++ + =a cbc bab acc b aTheo bt ng thc Nesbit chng minh bi 2 th: 23>+++++ b aca cbc ba Do ( )21232 2 2c b ac b aa cbc bab ac + +=|.|

\| + + >+++++ (pcm) Bi 4: Cho ba s thc dng a, b, c tha1 s + + c b a .Chng minh bt ng thc sau: 92121212 2 2>+++++ ab c ca b bc a Gii: Do1 s + + c b a ta c: ( )( )( ) ( ) ( ) | | 92121212 2 22121212 2 22121212121212 2 22 2 22 2 22 2 22 2 222 2 2>|.|

\|++++++ + + + + =|.|

\|++++++ + + + + =|.|

\|++++++ + >+++++ab c ca b bc aab c ac b bc aab c ca b bc aac bc ab c b aab c ca b bc ac b aab c ca b bc a 2.K thut i bin s Cnhngbitonvmtbiuthctonhctngicngknh,khnhnbit c phng hng gii. Bng cch i bin s, ta c th a bi ton v dngn gin v d nhn bit hn. Bi 1: Cho. , , , b CA a BC c AB ABC = = = ACMR: ( )( )( ) abc c b a b a c a c b s + + + (1) Gii:http://boxtailieu.netboxtailieu.net 11 t: +=+=+== += += +222y xcx zbz yaz c b ay b a cx a c b Khi bt ng thc (1) tng ng vi bt ng thc sau:

2.2.2. .x z z y y xz y x+ + +sDo trong tam gic, tng di ca hai cnh lun ln hn di cnh cn li nn : 0 , , > z y xp dng bt ng thc Cauchyta c: xyz zx yz xyx z z y y x= >+ + +.2.2.2

Hay( )( )( ) abc c b a b a c a c b s + + + (pcm) Bi 2: Cho. , , , b CA a BC c AB ABC = = = ACMR: 3 > ++ ++ + c b acb a cba c ba(1) Gii: t:

+=+=+=> = +> = +> = +222000y xcx zbz yaz c b ay b a cx a c b Khi v tri ca bt ng thc (1) tr thnh:

zy xyx zxz y2 2 2+++++ Ta c:

3 .22.22.22 2121212 2 2= + + >||.|

\|+ +|.|

\|+ +||.|

\|+ =+++++zyyzzxxzyxxyzyyzzxxzyxxyzy xyx zxz y Hay 3 > ++ ++ + c b acb a cba c ba(pcm) http://boxtailieu.netboxtailieu.net 12 Bi 3: Cho. , , , b CA a BC c AB ABC = = = ACMR: c b ac b acb a cba c ba+ + > ++ ++ +2 2 2(1) Gii:t: +=+=+=> = +> = +> = +222000y xcx zbz yaz c b ay b a cx a c b Khi bt ng thc (1) tng ng vi bt ng thc sau:

( ) ( ) ( )z y xzy xyx zxz y+ + >+++++4 4 42 2 2 Ta c: ( ) ( ) ( )y x zxyzzxyzxyyzxyzxxyzxyzzxyzxyyzxyzxxyzzxyyzxxyzzy xyx zxz y+ + = + + >|.|

\|+ +||.|

\|+ +||.|

\|+ = + + >+++++. . . 2121214 4 42 2 2Hayc b ac b acb a cba c ba+ + > ++ ++ +2 2 2(pcm) Bi 4: Cho 2, , , ,c b ap b CA a BC c AB ABC+ += = = = A.CMR: ( ) ( ) ( )( )( )( ) c p b p a ppc p b p a p >++2 2 21 1 1(1) Gii:Ta c: 02> += a c ba p Tng t:

00> > c pb p t: z y x pz c py b px a p+ + = > = > = > = 000 Khi bt ng thc (1) tng ng vi bt ng thc sau: http://boxtailieu.netboxtailieu.net 13

xyzz y xz y x+ +> + +2 2 21 1 1 Ta c: xyzz y xzx yz xy x z z y y xx z z y y x z y x+ += + + = + + >|.|

\|+ +||.|

\|+ +||.|

\|+ = + +1 1 1 1 1 1.1 1.11 121 1 121 1 121 1 1 12 2 2 2 2 22 2 2 2 2 2 2 2 2 Hay( ) ( ) ( )( )( )( ) c p b p a ppc p b p a p >++2 2 21 1 1(pcm) Bi 5: Cho ba s thc dnga, b, c.CMR:23>+++++ b aca cbc ba(1) Gii:t: += += +== += += +222z y xcy x zbx z yaz b ay a cx c b Khi bt ng thc (1) tr thnh:

212 2 2> ++ ++ +zz y xyy x zxx z y Ta c: 2323.22.22.22 232121212 2 2= + + >||.|

\|+ +|.|

\|+ +||.|

\|+ = ++ ++ +zyyzzxxzyxxyzyyzzxxzyxxyzz y xyy x zxx z y Hay23>+++++ b aca cbc ba (pcm) Bi 6: Cho 3 s thc khng ma, b, c tha( )( ) 1 = + + c b c a.CMR: ( ) ( ) ( )41 1 12 2 2>++++ c b c a b a(1) http://boxtailieu.netboxtailieu.net 14 Gii: t: = == = == += +y x b axyyxy x b axyy c bx c a111 Khi v tri ca bt ng thc (1) tr thnh:

( )41 1 12 2 2> + + y x y x Ta c: ( ) ( )( ) ( ) 4 2 2 .212 2 22121 1 1 1 12 22 22 22 22 22 22 22 2 2 2= + ++ > + + ++ =+ ++ = + += + +y xy xy xy xy xy xy xy xy x y x y x Vy( ) ( ) ( )41 1 12 2 2>++++ c b c a b a (pcm) Bi 7:Chox, y, zl cc s thc dng thay i v tha mn iu kin1 = xyz.Tm GTNN ca biu thc: ( ) ( ) ( )y y x xy x zx x z zx z yz z y yz y xA2 2 22 2 2+ +++ ++++= thi i hc khi A nm 2007 Gii:p dng bt ng thc Cauchy ta c: y y x xz zx x z zy yz z y yx xy y x xzxy z zx x z zyzx y yz z y yxyz x xy y x xxy zx x z zzx yz z y yyz xA222222

222222

22 .22 .22 .2 2 2 +++++>+++++>+++++> http://boxtailieu.netboxtailieu.net 15 t:( )( )( ) + =+ =+ + =+ =+ =+ =c b a z zc b a y yc b a x xy y x x cx x z z bz z y y a2 4914 2914 291222 Khi ( ) 2 3 12 692. . . 3 . . . 3 . 4 692

4 6922 4 4 2 4 2923 3= + + =||.|

\|+ + >((

|.|

\|+ + +|.|

\|+ + + >|.|

\| +++ ++ + >cbbaacbccaabcbbaacbccaabcc b abc b aac b aA

Du = xy ra1 = = = c b aVy GTNN ca A l2 3.K thut chn im ri im ri trong cc bt ng thc l gi tr t c ca bin khi du =trong bt ng thc xy ra. Trong cc bt ng thc du = thng xy ra cc trng hp sau: - Cc bin c gi tr bng nhau. Khi ta gi bi ton c cc tr t c ti tm - Khi cc bin c gi tr ti bin. Khi ta gi bi ton c cc tr t c ti bin Cn c vo iu kin xy ra ca du = trong bt ng thc ta xt cc k thut chn im ri trong cc trng hp trn 3.1K thut chn im ri trong bi ton cc tr xy ra bin Xt cc bi ton sau:Bi ton 1: Cho s thc2 > a . Tm gi tr nh nht (GTNN) ca 1aa A + =Sai lm thng gp l:21. 21= > + =aaaa A . Vy GTNN ca A l 2. Nguyn nhn sai lm: GTNN ca A l 21 a1= = aa v l v theo gi thuyt th 2 > a . Li gii ng:2542 . 3143 1.4243 141= + > + > + + = + =aaa aaaaa Ahttp://boxtailieu.netboxtailieu.net 16 Du = xy ra2 hay 14= = aaa Vy GTNN ca A l 25.V sao chng ta li bit phn tch c nh li gii trn. ychnh l k thut chn im ri trong bt ng thc. Quay li bi ton trn, d thy a cng tng th A cng tng. Ta d onA t GTNN khi2 = a . Khi ta ni A t GTNN ti im ri2 = a .Ta khng th p dng btngthcCauchychohaisa v 1avkhngthaquytcdu=.Vvyta phi tchahoc 1a khi p dng bt ng thc Cauchy th tha quy tc du =. GistasdngbtngthcCauchychocps |.|

\|aa 1,osaochotiimri 2 = a th aa 1=o, ta c s sau: 421 221 122 = = == = ooo oaaaKhi : aa aaa A14341+ + = + =v ta c li gii nh trn. Lu : gii bi ton trn, ngoi cch chn cp s |.|

\|aa 1,ota c th chn cc cc cp s sau: |.|

\|aa1, o hoc |.|

\|aa o,hoc |.|

\|aao1, . Bi ton 2: Cho s thc2 > a . Tm gi tr nh nht ca 12aa A + =S im ri: 841 241 1222= = == = ooo oaaahttp://boxtailieu.netboxtailieu.net 17 Sai lm thng gp l: 4982 . 72 . 21872187 1.8287 182 2= + > + = + > + + =aaaaa aaaA . Du = xy ra 2 = a . Vy GTNN ca A l 49 Nguyn nhn sai lm:Mc d GTNN ca A l 49 l p s ng nhngcch gii trn mc sai lm trong nh gi mu s: 2 . 21212 > >aal sai. Li gii ng: 4982 . 64386 1.8.8. 386 18 832 2= + > + > + + + =aaa a aaa aADu = xy ra2 = a Vy GTNN ca A l 49 Bi 1: Cho 2 s thc dnga, b tha1 s +b a . Tm GTNN ca 1abab A + =Phn tch: Ta c:4122s|.|

\| +sb aabS im ri: 161441414141= = == = ooo oabababGii: Ta c:

41 4122 > s|.|

\| +sabb aab 41741. 15 8 15116 2 15116 = > > + = ababab ababab ADu = xy ra 2141= = = b a abhttp://boxtailieu.netboxtailieu.net 18 Vy GTNN ca A l 417

Bi 2: Chos thc 6 > a . Tm GTNN ca 182aa A + =Phn tch: Ta c a aaaa A9 9 182 2+ + = + =D thy a cng tng th A cng tng. Ta d onA t GTNN khi6 = a . Ta c s im ri: 2423 362369 93662= = = == = ooo oaaaGii: Ta c:392436 . 23292423 9.9.2432423 9 924232 2 2= + >+ > + + + =aa aa aa aaA Du = xy ra69242= = aaa Vy GTNN ca A l 39Bi 3: Cho 3 s thc dnga, b, ctha20 3 2 > + + c b a . Tm GTNN ca

429 3c b ac b a A + + + + + =Phn tch: D onGTNN ca A t c khi20 3 2 = + + c b a,ti im ri4 , 3 , 2 = = = c b a . S im ri: 3423 223 322 = = == = ooo oaaahttp://boxtailieu.netboxtailieu.net 19 223 3232933 = = == = ||| |bbb 4 141444 = = == = cccGii: 13 5 2 3 343 2 4.4229.223.432432 444 292343= + + + >+ ++ + + >+ + +|.|

\|+ +|.|

\|+ +|.|

\|+ =c b accbbaac b accbbaaA Du = xy ra4 , 3 , 2 = = = c b aVy GTNN ca A l 13Bi 4: Cho3 s thc dnga, b, ctha >>812bcab. Chng minh rng:( )12121 8 1 1 12 > +|.|

\|+ + + + +abc ca bc abc b aPhn tch: D onGTNN ca A t c khi ==812bcab ,ti im ri2 , 4 , 3 = = = c b a . Gii: p dng bt ng thc Cauchy ta c: 12.6.9326 921 2.24.183224 1833= > + += > + +cac acac aabb aabb a 34 8.12.6.94812 6 943 2.8.16328 1643= > + + += > + +abcb c aabcb c abcc bbcc b http://boxtailieu.netboxtailieu.net 20 4138 .2413.481322413.481322413481331312 .2413.181322413.1813224131813= > > += > > +c b c bb a b a Cng theo v cc bt ng thc trn ta c: ( )12121 8 1 1 12 > +|.|

\|+ + + + +abc ca bc abc b a (pcm) 3.2K thut chn im ri trong bi ton cc tr t c ti tm Xt bi ton sau: Bi ton:Cho 2 s thc dnga, b tha1 s +b a .. Tm GTNN cab ab a A1 1+ + + =Sai lm thng gp l:41.1. . 41 14= > + + + =b ab ab ab a A Vy GTNN ca A l 4. Nguyn nhn sai lm: GTNN ca A l 41 a1 1= = = = = bb ab a . Khi 1 2 > = +b atrigi thuyt . Phn tch: Do A l biu thc i xng via, b nn ta d on GTNN ca A t ti

21= = b aS im ri: 4122121 12121= = = == = = = ooo o ob ab ab aLi gii ng:( ) 5 3 8 31.1. 4 .. 4 4 3 31 14 44= > + > |.|

\|+ + + = b ab ab a b ab ab a ADu = xy ra 21= = b aVy GTNN ca A l5 http://boxtailieu.netboxtailieu.net 21 Bi 1:Cho 3 s thc dnga, b, ctha 23s + + c b a . Tm GTNN cac b ac b a A1 1 1+ + + + + =Phn tch: Do A l biu thc i xng via, b, cnn ta d on GTNN ca A t ti

21= = = c b aS im ri: 4122121 1 12121= = = = == = = = = = ooo o o oc b ac b ac b aGii: ( )213291231.1.1. 4 . 4 . 4 63 3 31 1 14 4 46= >+ + > |.|

\|+ + + + + =c b ac b ac b ac b ac b ac b a A Du = xy ra21= = = c b a Vy GTNN ca A l 213

Bi 2:Cho 3 s thc dnga, b, ctha 23s + + c b a . Tm GTNN ca c b ac b a A1 1 12 2 2+ + + + + =Phn tch: Do A l biu thc i xng vi a, b, cnn ta d on GTNN ca A t ti

21= = = c b aS im ri: http://boxtailieu.netboxtailieu.net 22 82412 1 1 141212 2 2= = = = == = = = = = ooo o o o c b ac b ac b aGii: 4272 .494931.4949 1. 9491 1 14381.81.81.81.81.81. . . 9434343818181818181392 2 22 2 2= + >+ ++ > + >|.|

\|+ + + >+ + +|.|

\|+ + + + + + + + =c b aabcc b a c b a c b ac b ac b a c b a c b ac b a A Du = xy ra21= = = c b a Vy GTNN caAl 427

Bi 3:Cho 2 s thc dnga, b. Tm GTNN ca b aababb aA+++=Phn tch: Do A l biu thc i xng via, bnn ta d on GTNN ca A t tib a =S im ri: 421 22122 2= = = =+= =+ = ooo ooaab aabaaabb ab aGii:

( )2523142 . 3.4243 4= + = +++>++||.|

\|+++=ababb aababb aabb ab aababb aADu = xy ra b a = Vy GTNN ca A l 25

Bi 4:Cho 3 s thc dnga, b, c. Tm GTNN cacb aba cac bb aca cbc baA+++++++++++=Phn tch: http://boxtailieu.netboxtailieu.net 23 Do A l biu thc i xng vi a, b, cnn ta d on GTNN ca A t tic b a = =S im ri: 4221221= = =+=+=+=+=+=+ = = ooo o o o cb aba cac bb aca cbc bac b aGii:

|.|

\|+ + + + + ++ + ++ + +>|.|

\| ++++++|.|

\| +++++++++++=cbcababcacabcb aba cac bb aca cbc bacb aba cac bcb aba cac bb aca cbc baA 434.4.4. . . 6 434 4 46

215293 . . . . . . 6 .4336= + = + >cbcababcacab Du = xy rac b a = = Vy GTNN ca A l 215

Bi 5:Cho 2 s thc dnga, b tha1 s +b a . Tm GTNN ca : ab b aA21 12 2++=Phn tch: Do A l biu thc i xng vi a, bnn ta d on GTNN ca A t ti

21= = b aS im ri: 1 2 22221212 2= = ==+ = = o oooabb ab aGii: ( ) ( )44221. 221221 12 2 2 2 2 2 2>+=+ +>+> ++=b a ab b a ab b a ab b aADu = xy ra 21122 2= = = += + b ab aab b a http://boxtailieu.netboxtailieu.net 24 Vy GTNN ca A l 4 Bi 6:Cho 2 s thc dng a, btha1 s +b a . Tm GTNN caab b aA21 112 2++ +=Phn tch: Do A l biu thc i xng vi a, bnn ta d on GTNN ca A t ti

21= = b aS im ri: 32322213211212 2= = ==+ + = = ooo oabb ab aGii: ( )( )abab b aab ab b aab ab b aab ab b aA314 143126 11. 231 6 1123161 112 2 22 22 2++ + += ++ + +>++ +>+ ++ +=

( )||.|

\||.|

\| +s|.|

\| ++|.|

\| ++ + +> 2Do 23124 1422 22b aabb a b ab a ( ) ( ) 341 242 2b a b a +++ +>

381 . 341 1 . 24= ++>Du = xy ra 2116 12 2= = = +== + + b ab ab aab b a Vy GTNN ca A l 38 http://boxtailieu.netboxtailieu.net 25 Bi 7:Cho 2 s thc dnga, b tha1 s +b a . Tm GTNN caabab b aA 41 12 2+ ++=Phn tch: Do A l biu thc i xng vi a, bnn ta d on GTNN ca A t ti

21= = b aS im ri: 2424 121212 2= = ==+ = = ooo oabb ab a441 4 11 421= = == = = ||| |ababb aGii: ( )( )abb aab ab b aab ababab b aab ababab b aA4124412221. 24141. 4 22124141421 12 2 22 22 2+ ++= + ++ +>+ ++>+ + + ++=

( )||.|

\||.|

\| +s|.|

\| ++ ++> 2Do 2412422 2b aabb ab a ( )7 215252= + >++>b a Du = xy ra 21141422 2= = = +=== + b ab ab aababab b a Vy GTNN ca A l 7 http://boxtailieu.netboxtailieu.net 26 Bi 8:Cho 2 s thc dnga, b tha1 s +b a . Tm GTNN ca 2 2 3 31 1 1ab b a b aA + ++=Phn tch: Do A l biu thc i xng vi a, bnn ta d on GTNN ca A t ti

21= = b aS im ri: 2424 1 121212 23 3= = = ==+ = = ooo o o ab b ab ab a Gii: 52 2 2 21521.21.21.21.1521212121 12 2 2 2 3 352 2 2 2 3 32 2 2 2 3 3ab b a ab b a b aab b a ab b a b aab b a ab b a b aA+ + + + +>+>+ + + ++=

( ) ) (253b a ab b a + + +>( )2041125

2Do 4) (25

233=+>||.|

\||.|

\| +s++ +>b aabb ab a Du = xy ra 2112121 12 2 3 3= = = +== =+ b ab ab aab b a b a Vy GTNN ca A l 20 http://boxtailieu.netboxtailieu.net 27 Bi 9:Cho ba s thc dngz y x , ,tha41 1 1= + +z y x. Tm GTLN ca z y x z y x z y xP2121 21+ +++ +++ += thi i hc khi A nm 2005 Gii:

||.|

\|+ + + s = s+ + +=+ + z y x x z y x xz y x xz y x x z y x1 1 1 1161 1.1.1.141. . . 41 12144 Tng t: ||.|

\|+ + + s+ + z y y x z y x1 1 1 116121 ||.|

\|+ + + s+ + z z y x z y x1 1 1 116121

Cng theo v 3 bt ng thc trn, ta c: 14 4 41612121 21=||.|

\|+ + s+ +++ +++ +=z y x z y x z y x z y xPDu = xy ra 4334 1 1 1= = = = = = z y xz y x Vy GTLN ca P l 1 4.K thut nhn thm h s Bi 1: Tm GTLN ca :( ) ( ) 1 , 0 , 12e = a -a a AGii:Do0 1 , > -a a nn p dng bt ng thc Cauchy ta c: ( ) ( ) 274278.2132 2212 2212 22132s =|.|

\| + +s = =A a - a aa - a.a a - a A Du = xy ra 322 2 = = a aVy GTLN ca A l 274 http://boxtailieu.netboxtailieu.net 28 Bi 2: Tm GTLN ca :( ) ( ) 2 , 0 , 23e = a -a a AGii:p dng bt ng thc Cauchy ta c: ( )162743 6313 6 . . .314=|.|

\| + + +s =a a a aa a a a ADu = xy ra 233 6 = = a aVy GTLN ca A l 1627 Bi 3:Cho cc s thc dnga, btha ss43ba. Tm GTLN ca( )( )( ) b a b a A 3 2 4 3 + =Gii:p dng bt ng thc Cauchy ta c: ( )( )( ) 3633 2 3 12 2 6613 2 3 12 2 6613=|.|

\| + + + s + =b a b ab a b a ADu = xy ra == = + = = 206 3 2 3 12 2 6bab a b aVy GTLN ca A l 36 Bi 4:Cho cc s thca, b, c tha >>>1262cba. Tm GTLN ca:abcc ab b ca a bcA4 312 6 2 + + =Gii:p dng bt ng thc Cauchy ta c: ( )( )( )( )( )( )2 8 64 444 4 4 12.644 . 4 . 4 . 1264129 333 3 6.93 . 3 . 6962 222 2.22 . 2224 44443 3333abc abc c abcabc ababc b cabcab caabc a bcabca bc= =+ + + s = =+ + s = =+ s = Khi ta c: http://boxtailieu.netboxtailieu.net 29 3 34 39 312 852 819 312 21 12 6 2+ = + + s + + =abcc ab b ca a bcADu = xy ra ==== = = 16944 123 62 2cbacba Vy GTLN ca A l 39 312 85+Bi 5: Cho 3 s thc dnga, b, ctha1 = + + c b a .Tm GTLN ca:a c c b b a A + + + + + =Phn tch: Do A l biu thc i xng via, b, cnn ta d on GTNN ca A t ti = += += + = = =323232

31a cc bb ac b aGii:p dng bt ng thc Cauchy ta c: ( )( )( )( )(3)(2)(1)

232.23

232.23

232.2332.23+ +s ++ +s ++ +s + = +a ca cc bc bb ab a b a Cng theo v cc bt ng thc (1), (2) v (3) ta c: ( )6232. 3 2.23=+ + +s + + + + + =c b aa c c b b a ADu = xy ra 31323232= = = = += += + c b aa cc bb a http://boxtailieu.netboxtailieu.net 30 Vy GTLN ca A l6Lu : Trong bi ton s dng k thut nhn thm h s, ta s s dng k thut chn im ri tm h s cho ph hp. Bi 6: Cho 3 s thc dnga, b, ctha3 = + + c b a .Chng minh rng:3 3 3 33 3 2 2 2 s + + + + + a c c b b a Phn tch: Do biu thc cho l biu thc i xng via, b, cnn ta d on du = xy ra khi: = += += + = = =3 23 23 2 1a cc bb ac b aGii:p dng bt ng thc Cauchy ta c: ( )( )(3)(2)(1)

9 32 62

9 32 62

9 32 633 3 2913 . 3 . 291233333 3333a ca c c bc bb a b ab a b a+ +s ++ +s ++ +=+ + +s + = + Cng theo v cc bt ng thc (1), (2) v (3) ta c: ( )333 3 33 39 33 182 2 2 =+ + +s + + + + +c b aa c c b b a(pcm) Bi 7: Choa, b, c| | 2 ; 2 etha3 = + + c b a .Chng minh rng:3 3 4 4 42 2 2s + + c b aPhn tch: Do biu thc cho l biu thc i xng vi a, b, cnn ta d on du = xy ra khi: = = = = = =3 43 43 4 1222cbac b aGii:http://boxtailieu.netboxtailieu.net 31 p dng bt ng thc Cauchy ta c: ( )( )(3)(2)(1) 3 274

3 274

3 2723 4.313 431422222 22 2ccbba aa as s =+ s = Cng theo v cc bt ng thc (1), (2) v (3) ta c: ( )3 2214 4 42 2 22 2 2c b ac b a+ + s + + M theo bt ng thc Bunyakovski ta c( ) ( )( )( )31 1 122 2 22 2 2 2c b ac b ac b a c b a+ +> + + + + + + s + + nn( )3 33 23214 4 422 2 2=+ +s + + c b ac b a(pcm) 5.K thut h bc 5.1 Bi ton 1Choccsthcdnga, b, cthamniukin1 = + + c b a (*).Tmgitr nh nht ca biu thc 2 2 2c b a A + + =Phntch:Schnhlchvsmcaccbiuthc 2 2 2c b a + + vc b a + + gi cho ta s dng bt ng thc Cauchy h bc 2 2 2c b a + + .Nhng ta cn p dng cho baonhius v l nhng s no? Cn c vo bc ca ccbin sa,b, ctrong ccbiuthctrn(sbcgim2ln)thtacnpdngbtngthcCauchyln ltcho 2 2 , b a v 2c cngvi1hngsdngtngngkhclmxuthin b a,vc . Do a, b, c dng v c vai tr nh nhau nn ta d on A t gi tr nh nht khic b a = = , t (*) ta c 31= = = c b a . Mt khc th du = ca bt ng thc Cauchy xy rakhi chkhi cc s tham gia bng nhau. Khi ta c li gii nh sau: Li gii:p dng bt ng thc Cauchy cho 2 s: 2a v 91 ta c: http://boxtailieu.netboxtailieu.net 32 a a a3291. 2912 2= > + (1)Du = xy ra 31912= = a aTng t: b b32912> +(2)Du = xy ra 31= bc c32912> +(3)Du = xy ra 31= cCng theo v cc bt ng thc (1), (2)v (3) ta c: ( )313232312 2 2 2 2 2> + + = + + > + + + c b a c b a c b a . Du = xy ra 31= = = c b aVy GTNN ca A l 31 Bi 1: Cho cc s thc dng a, b tha mn iu kin13 3= +b a (*). Tm gi tr ln nht ca biu thcb a A + =Phntch:Cncvobccaccbinsa,btrongccbiuthctrn(sbc gim 6 ln) th ta cn p dng bt ng thc Cauchy ln lt cho3av 3bcng vi 5 hng s dng tng ng khc lm xut hinavb . Do a, b dng v c vaitrnhnhaunntadonAtgitrlnnhtkhib a = ,t(*)tac 213 3= = b a . Mt khc th du = ca bt ng thc Cauchy xy rakhi chkhi cc s tham gia bng nhau. Khi ta c li gii nh sau: Gii:p dng bt ng thc Cauchy cho 6 s: 3av 5 s 21 ta c: a a a .21. 621. . 621. 56 5653 3=|.|

\|> + (1)Du = xy ra 332121= = a aTng t: b b b .21. 621. . 621. 56 5653 3=|.|

\|> + (2)Du = xy ra 321= bCng theo v cc bt ng thc (1) v (2) ta c: ( ) ( )6 56 5 6 53 3221. 6 5 121. 6 5 s + + > + + > + + b a b a b a b ahttp://boxtailieu.netboxtailieu.net 33 Du = xy ra 321= = b aVy gi tr ln nht ca A l 6 52Bi 2:Cho 3 s thc dnga, b, c tha3 = + + ca bc ab . CMR:33 3 3> + + c b aGii: p dng bt ng thc Cauchy ta c: ab b a b a 3 3 13 3 3 3 3= > + + (1) ;bc c b 3 13 3> + + (2) ; ca a c 3 13 3> + + (3) Cng theo v cc bt ng thc (1), (2) v (3) ta c: ( ) ( )( ) 3 . 3 3 23 3 23 3 33 3 3> + + + + + > + + +c b aca bc ab c b a 33 3 3> + + c b a(pcm)Bi3:Cho 3 s thc dnga, b, c tha33 3 3= + + c b a . CMR:35 5 5> + + c b aGii: p dng bt ng thc Cauchy cho 5 s: 3 s 5av 2 s 1, ta c: 3 5 15 55 1 . 1 5 2 3 a a a = > +(1) Tng t: 3 55 2 3 b b > +(2) ;3 55 2 3 c c > +(3) Cng theo v cc bt ng thc (1), (2) v (3) ta c: ( ) ( )( ) 3 . 5 6 35 6 35 5 53 3 3 5 5 5> + + + + + > + + +c b ac b a c b a 35 5 5> + + c b a (pcm) Bi 4: Cho 3 s thc dnga, b, c tha33 3 3 3 3 3= + + a c c b b a . CMR: 37 7 7> + + c b aGii: p dng bt ng thc Cauchy cho 7 s: 3 s 7a, 3s 7bvs 1, ta c: 3 3 7 21 21 7 77 1 . 7 1 3 3 b a b a b a = > + +(1) Tng t: 3 3 7 77 1 3 3 c b c b > + +(2) ;3 3 7 77 1 3 3 a c a c > + +(3) Cng theo v cc bt ng thc (1), (2) v (3) ta c: ( ) ( )( ) 3 . 7 3 67 3 67 7 73 3 3 3 3 3 7 7 7> + + + + + > + + +c b aa c c b b a c b a http://boxtailieu.netboxtailieu.net 34 37 7 7> + + c b a(pcm) Bi 5:Cho 2 s thc dnga, b. CMR:ab b a b a + + > + + 2 2 42 2 Gii: p dng bt ng thc Cauchy, ta c: a a a 4 4 . 2 42 2= > + (1); b b 4 42> +(2) ;ab b a 22 2> +(3) Cng theo v cc bt ng thc (1), (2) v (3) ta c: ab b a b a 2 4 4 8 2 22 2+ + > + +ab b a b a + + > + + 2 2 42 2 (pcm) Bi 6:Cho 3 s thc dnga, b, c. CMR:ab c ca b bc a c b a2 2 2 3 3 3+ + > + +Gii: p dng bt ng thc Cauchy cho 6 s: 4 s 3a ,1 s 3bv 1 s 3c ta c: bc a c b a c b a2 6 3 3 12 3 3 36 . . 6 4 = > + +(1) Tng t:ca b a c b2 3 3 36 4 > + +(2) ;ab c b a c2 3 3 36 4 > + +(3) Cng theo v cc bt ng thc (1), (2) v (3) ta c: ( ) ( ) ab c ca b bc a c b a2 2 2 3 3 36 6 + + > + +ab c ca b bc a c b a2 2 2 3 3 3+ + > + + (pcm) Bi 7:Cho cc s thc dnga, b, c, m, n. CMR:n m n m n m n m n m n ma c c b b a c b a + + > + ++ + + Gii: p dng bt ng thc Cauchy cho m+n s: m s n ma+ v n s n mb+ ta c: ( ) ( ) ( ) ( )n m n mnn mmn m n m n mb a n m b a n m nb ma . . + = + > ++ + + + + (1) Tng t:( )n m n m n mc b n m nc mb . + > ++ +(2) ( )n m n m n ma c n m na mc . + > ++ + (3) Cng theo v cc bt ng thc (1), (2) v (3) ta c: ( )( ) ( )( )n m n m n m n m n m n ma c c b b a n m c b a n m + + + > + + ++ + + n m n m n m n m n m n ma c c b b a c b a + + > + ++ + + (pcm) Lu : Bt ng thc chng ta va chng minh s c s dng trong chng minh cc bi ton sau ny. http://boxtailieu.netboxtailieu.net 35 Bi 8:Cho 3 s thc dnga, b, c tha1 = abc. Chng minh bt ng thc sau: 11111113 3 3 3 3 3s+ +++ +++ + a c c b b a Gii: T kt qu bi 7 ta c n m n m n m n m n m n ma c c b b a c b a + + > + ++ + + Chn ===a cnm12ta c: ( ) (1)1 do11112 2 2 2 3 32 2 3 33 2 2 2 2 2 3 3 3=+ +=+ +=+ +s+ ++ > + + + = + + > + +abcc b acabc a b b aabca b b a b aa b b a b aa a b b a a a a b b a a b a Tng t:c b aac b + +s+ + 113 3 (2) c b aba c + +s+ + 113 3 (3) Cng theo v cc bt ng thc (1), (2) v (3) ta c: 11111113 3 3 3 3 3=+ ++ +s+ +++ +++ + c b ac b aa c c b b a(pcm) 5.2Bi ton 2 Cho cc s thc dnga, b, ctha mn iu kin1 = + + ca bc ab . Chng minh rng:4 10 102 2 2> + + c b aGii: p dngbt ng thc Cauchy ta c: accaca 42. 8 2282222= > +bccbcb 42. 8 2282222= > +ab b a b a 4 2 . 2 2 2 22 2 2 2= > +Cng theo v 3 bt ng thc trn, ta c: ( ) 4 1 . 4 4 10 102 2 2= = + + > + + ca bc ab c b ahttp://boxtailieu.netboxtailieu.net 36 Du = xy ra == ====34312 228282 22222cb ab acbca y l mt li gii ngn gn nhng c v hi thiu t nhin. Chng ta s thc mc ti saolitchc2 8 10 + = .Nutchcchkhc,chnghn4 6 10 + = liucgii ckhng?Ttnhinmicchtchkhcukhngdnnktqu,vtch 2 8 10 + =cng khng phi l s may mn. By gi ta s tm l do vic tch2 8 10 + = bi ton trn. Vi 10 0 < +bccbcb o o o 22. 222222= > +( ) ( ) ( ) ( ) ( )ab b a b a o o o o o 2 20 10 10 2 10 102 2 2 2 = > + Cng theo v 3 bt ng thc trn, ta c: ( ) ( )ab bc ac c b a o o 2 20 2 10 102 2 2 + + > + +Lc ny ta cn bng iu kin gi thuyt, tc l:

> == = + + = =1022580 200 41 2 4 80 400 2 2 20 22 2ooo o o o o o o8 = oKhi ta c li gii bi ton nh trn. Bi 1:Cho cc s thc dnga, b, ctha mn iu kin5 = + + ca bc ab . CMR: : 10 3 32 2 2> + + c b aGii: p dngbt ng thc Cauchy ta c: accaca 22. 2 2222222= > +bccbcb 22. 2 2222222= > +http://boxtailieu.netboxtailieu.net 37 ab b a b a 2 . 22 2 2 2= > +Cng theo v 3 bt ng thc trn, ta c: ( ) 10 5 . 2 2 3 32 2 2= = + + > + + ca bc ab c b a5.3Bi ton 3 Cho cc s thc dng a, b tha mn iu kin13 3s +b a . Tm gi tr ln nht ca biu thcb a A 4 + =Phn tch: D on A t GTLN khi13 3= +b a Gi s A t GTLN khi ==|oba . Ta c13 3= + | o(1) p dng bt ng thc Cauchy cho 3 s: 3av 2 s 3ota c: ( ) a a a2 323 3 3 33 . . 3 2 o o o = > +Tng t: ( ) b b b2 323 3 3 33 . 3 2 | | | = > +Cng theo v cc bt ng thc trn ta c: ( ) ( ) b a b a2 2 3 3 3 33 3 2 | o | o + > + + + xut hin v phib a 4 +ta chn| o,sao cho( ) 221414 : 3 : 3222 2= = =|o|o| o b a b a T (1) v (2) ta c h: === +=33 233121333 3|o| o|o Khi ta c li gii sau: Gii: p dng bt ng thc Cauchy ta c: a a a333 33191.91. . 39191= > + +b b33349898> + +Cng theo v cc bt ng thc trn ta c: http://boxtailieu.netboxtailieu.net 38 ( ) ( )( ) | |3 3 3 333 33 3 2 3 44312s + + s + + > + +b a b ab a b a Du = xy ra khi ====33 23398913333baba Vy GTLN ca A l 33 3 Bi 1:Cho cc s thc dnga, b, ctha mn iu kin3 = + + c b a . Tm GTNN ca2 2 23 6 4 c b a A + + =Phn tch: Vi 0 , , > | o . p dngbt ng thc Cauchy ta c: a a a o o o 4 2 . 4 2 42 2= > +b b b | | | 6 2 . 6 2 62 2= > +c c c 3 2 . 3 2 32 2= > +Cng theo v 3 bt ng thc trn, ta c: c b a c b a | o | o 3 2 6 2 4 2 3 6 42 2 2+ + > + + + + + +Du = xy ra 33 6 436433643222= + + ==== + +==== + + | o|o|ocbac b acbac b a Chn | o , , sao cho | o 3 6 4 = =Ta c h phng trnh: 3 6 433 6 4= == + + | o | o http://boxtailieu.netboxtailieu.net 39 == = =|.|

\|+ + = + + === + +316384332312133 . 346 . 644346433 6 4|ooo o ooo| | o Khi ta c li gii bi ton nh sau Gii:p dngbt ng thc Cauchy ta c: a a a 8 4 . 4 2 4 42 2= > +b b b 836. 8 23862 2= > +c c c 8316. 3 231632 2= > +Cng theo v 3 bt ng thc trn, ta c: ( )12 3 6 424 8316384 3 6 42 2 22 2 2> + + = + + > + + + + +c b ac b a c b a Du = xy ra ======= + +3432131633864 43222cbacbac b a VyGTNN ca A l 12 6.K thut cng thm Bi 1:Cho 3 s thc dnga, b, c. Chng minh bt ng thc sau: c b a accbba 1 1 12 2 2+ + > + +Gii:p dng bt ng thc Cauchy ta c: http://boxtailieu.netboxtailieu.net 40

b a baa ba 2 1. 212 2= > +(1) ;c b cb 2 12> +(2); a c ac 2 12> +(3)Cng theo v cc bt ng thc (1), (2) v (3) ta c: c b a c b a accbba 2 2 2 1 1 12 2 2+ + > + + + + +c b a accbba 1 1 12 2 2+ + > + + (pcm) Bi 2: Cho 3 s thc dnga, b, c. Chng minh bt ng thc sau: 3 2 2 22 2 2c b ab aca cbc ba + +>+++++ Gii:p dng bt ng thc Cauchy ta c: 3292.229222 2a c bc ba c bc ba=++>+++ (1); 329222b a ca cb>+++ (2) ;329222c b ab ac>+++(3) Cng theo v cc bt ng thc (1), (2) v (3) ta c: ( ) ( )32932 2 22 2 2c b a c b ab aca cbc ba + +>+ +++++++ 3 2 2 22 2 2c b ab aca cbc ba + +>+++++(pcm) Lu :Trong bi ton s dng k thut cngthm h s, ta s s dng k thut chn im ri v k thut h bc tm hng t cho ph hp. V d:-i vibi 1bt ng thc cho c tnh i xng via, b, c nn ta d on du = xy ra khic b a = = . Khi 12 2a aaba= = , ta chn 1a. -i vibi 2bt ng thc cho c tnh i xng via, b, c nn ta d on du=xyrakhic b a = = .Khi 3 2 22 2aa aac ba=+=+,munsdngbtng thcCauchylmmtmuthtacngthm 92 c b +.Chnmuls9v 3 9292 a a a c b=+=+. http://boxtailieu.netboxtailieu.net 41 Bi 3: Cho 3 s thc dnga, b, c. Chng minh bt ng thc sau: ( ) c b acaa cbcc babb a+ + >+++++23 3 3 3 3 3 Gii:Ta c: caacbccbabbacaa cbcc babb a2 2 2 2 2 2 3 3 3 3 3 3+ + + + + =+++++ p dng bt ng thc Cauchy ta c: a bbabba2 . 22 2= > +(1); b aab22> + (2);b ccb22> +(3) ; c bbc22> + (4) ;c aac22> +(5); a cca22> + (6)Cng theo v cc bt ng thc t (1) n (6) ta c: ( ) ( )( ) c b acaacbccbabbac b a c b acaacbccbabba+ + > + + + + + + + > + + + + + + + +24 22 2 2 2 2 22 2 2 2 2 2 ( ) c b acaa cbcc babb a+ + >+++++ 23 3 3 3 3 3 (pcm) Bi 4:Cho 3 s thc dnga, b, c. Chng minh bt ng thc sau: c b a accbba 1 1 1323232+ + > + +Gii:p dng bt ng thc Cauchy ta c:

b a a baa a ba 3 1.1. 31 133232= > + +(1) ;c b b cb 3 1 132> + + (2); a c c ac 3 1 132> + +(3)Cng theo v cc bt ng thc (1), (2) v (3) ta c: |.|

\|+ + >|.|

\|+ + + + +c b a c b a accbba 1 1 131 1 12323232 c b a accbba 1 1 1323232+ + > + + (pcm) http://boxtailieu.netboxtailieu.net 42 Bi 5:Cho 3 s thc dnga, b, c. Chng minh bt ng thc sau: 2 2 23 3 3c b aaccbba+ + > + +Gii:p dng bt ng thc Cauchy ta c: 2323 323 33 . . 3 a bbababbaba= > + +(1); 2 23 33b ccbcb> + +(2);2 23 33c aacac> + + (3) Cng theo v cc bt ng thc t (1), (2) v (3) ta c: ( ) ( )2 2 2 2 2 23 3 33 2 c b a c b aaccbba+ + > + + +||.|

\|+ +2 2 23 3 3c b aaccbba+ + > + + (pcm) Bi 6: Cho 3 s thc dnga, b, c tha1 = abc . Chng minh bt ng thc sau:( )( ) ( )( ) ( )( ) 431 1 1 1 1 13 3 3>+ +++ +++ + b aca cbc ba Gii:p dng bt ng thc Cauchy ta c: ( )( ) ( )( )ac bc ba c bc ba4381.81.1 1381811 133 3=+ ++ +>+++++ + (1); ( )( )ba ca cb4381811 13>+++++ + (2); ( )( )cb ab ac4381811 13>+++++ + (3) Cng theo v cc bt ng thc (1), (2) v (3) ta c: ( )( ) ( )( ) ( )( )( ) ( )( )( ) ( )( ) ( )( )( )43432343211 1 1 1 1 14343411 1 1 1 1 133 3 33 3 3= > + + >+ +++ +++ ++ + > + + + ++ +++ +++ +abc c b ab aca cbc bac b a c b ab aca cbc ba

(pcm) http://boxtailieu.netboxtailieu.net 43 Bi 7:Cho 3 s thc dnga, b, c. Chng minh bt ng thc sau: c b aabccabbca+ + > + +242424 Gii:p dng bt ng thc Cauchy ta c: a c c bbcac c bbca4 . . . 442424= > + + +(1) b a a ccab424> + + +(2) c b b aabc424> + + +(3)Cng theo v cc bt ng thc (1), (2) v (3) ta c: ( ) ( ) c b a c b aabccabbca+ + > + + + + + 4 3242424 c b aabccabbca+ + > + + 242424 (pcm) Bi 8: Cho 3 s thc dnga, b, c. Chng minh bt ng thc sau: ( ) ( ) ( )|.|

\|+ + >+++++ c b a a c bcac b abcb a cab 1 1 1212 2 2 Gii:p dng bt ng thc Cauchy ta c: ( ) ( ) c abb ab a cababb ab a cab 14. 242 2=++>+++(1) ( ) a bcc bc b abc 142>+++ (2) ; ( ) b caa ca c bca 142>+++ (3) Cng theo v cc bt ng thc (1), (2) v (3) ta c: ( ) ( ) ( )( ) ( ) ( ) c b a c a b c a b a c bcac b abcb a cabc b a caa cbcc babb aa c bcac b abcb a cab1 1 141414141 41411 1 14 4 42 2 22 2 2+ + > + + + + + +++++++ + >+++++++++++( ) ( ) ( )|.|

\|+ + >+++++c b a a c bcac b abcb a cab 1 1 1212 2 2 (pcm) http://boxtailieu.netboxtailieu.net 44 Bi 9: Cho 3 s thc dnga, b, c tha32 2 2= + + c b a . Chng minh rng: 233 3 3>+++++ b aca cbc ba Gii:p dng bt ng thc Cauchy ta c: ( ) ( )23 34. 24ac b ac ba c b ac ba=++>+++ (1) ; ( )234ba c ba cb>+++ (2) ; ( )234cb a cb ac>+++ (3) Cng theo v cc bt ng thc (1), (2) v (3) ta c: ) (1' 22 2 23 3 3c b aca bc abb aca cbc ba+ + >+ +++++++ Mt khc ta c:n m n m n m n m n m n ma c c b b a c b a + + > + ++ + + Chn ==11nmta c: ) (2'2 2 2 2 22 2 2ca bc ab c b aca bc ab c b a+ +>+ ++ + > + + Cng theo v cc bt ng thc (1)v (2) ta c: 2 2 22 2 22 2 2 3 3 3ca bc abc b ac b a ca bc abb aca cbc ba + ++ + + >+ +++ +++++++232 2 2 2 3 3 3=+ +>+++++c b ab aca cbc ba (pcm) Bi 10: Cho 3 s thc dnga, b, c. Chng minh bt ng thc sau: 3 3 3252525c b aaccbba+ + > + +Gii:p dng bt ng thc Cauchy ta c: 3 2252252 . 2 a abbaabba= > +(1) ; 3 2252b bccb> +(2) ; 3 2252c caac> +(3) Cng theo v cc bt ng thc (1), (2) v (3) ta c: http://boxtailieu.netboxtailieu.net 45 ( ) ) (1' 23 3 3 2 2 2252525c b a ca bc abaccbba+ + > + + + + +Mt khc ta c:n m n m n m n m n m n ma c c b b a c b a + + > + ++ + + Chn ==21nmta c: ) (2' 2 2 2 3 3 3ca bc ab c b a + + > + +Cng theo v cc bt ng thc (1)v (2) ta c: ( )2 2 2 3 3 3 3 3 3 2 2 22525252 ca bc ab c b a c b a ca bc abaccbba+ + + + + > + + + + + + + +3 3 3252525c b aaccbba+ + > + + (pcm) Bi 11: Cho 3 s thc dnga, b, c. Chng minh bt ng thc sau: ( )2 2 23 3 3312 2 2c b aa ccc bbb aa+ + >+++++ Gii:p dng bt ng thc Cauchy ta c: ( ) ( )23 33292.22922ab a ab aa b a ab aa=++>+++ (1) ; ( )2332922bc b bc bb>+++ (2) ; ( )2332922cb c cb cc>+++ (3) Cng theo v cc bt ng thc (1), (2) v (3) ta c: ( ) ( ) ( )( ) ( ) ) (1'95922 2 23292912 2 22 2 23 3 32 2 2 2 2 23 3 3c b a ca bc aba ccc bbb aac b a ca bc ab c b aa ccc bbb aa+ + > + + +++++++ + > + + + + + ++++++ Mt khc ta c:n m n m n m n m n m n ma c c b b a c b a + + > + ++ + + Chn ==11nmta c: ( ) ( ) ) (2'92 92 2 2 22 2 2ca bc ab c b aca bc ab c b a+ + > + + + + > + + Cng theo v cc bt ng thc (1)v (2) ta c: ( ) ( ) ( ) ( ) ca bc ab c b a c b a ca bc aba ccc bbb aa+ + + + + > + + + + + ++++++ 929592922 2 22 2 2 2 2 23 3 3 http://boxtailieu.netboxtailieu.net 46 ( )2 2 23 3 3312 2 2c b aa ccc bbb aa+ + >+++++(pcm) Bi 12: Cho 3 s thc dnga, b, c. Chng minh bt ng thc sau: c b a cb aba cac b 2 2 22 2 2+ + >+++++ Gii:p dng bt ng thc Cauchy ta c:

4 4. 242 2a c b ac bc b ac b=++>+++(1); 4 42b a c ba c>+++(2);4 42c b a cb a>+++(3) Cng theo v cc bt ng thc (1), (2) v (3) ta c: ) (1'4 4 4 4 4 42 2 2c b a a c c b b a cb aba cac b+ + >+++++++++++ M ta c: ) (2'424 1.121 1b aabb a b a +> = > +; ) (3' 4 1 1c b c b +> + ;) (4' 4 1 1a c a c +> +Cng theo v cc bt ng thc (1), (2), (3) v (4) ta c: 4 4 4

4 4 4 2 2 2 4 4 42 2 2a c c b b a c b a c b a a c c b b a cb aba cac b++++++ + + > + + ++++++++++++c b a cb aba cac b 2 2 22 2 2+ + >+++++(pcm) Bi 13: Cho 3 s thc dnga, b, c. Chng minh bt ng thc sau: b aaccbba342 2 2+ > + + Du = ca bt ng thc xy ra khi2c b a = =Gii:p dng bt ng thc Cauchy ta c: a bbabba2 . 22 2= > +(1); b ccb4 42> +(2) ; c aac442> +(3) Cng theo v cc bt ng thc t (1), (2) v (3) ta c: http://boxtailieu.netboxtailieu.net 47 c b a c b aaccbba4 4 2 442 2 2+ + > + + + + +b aaccbba342 2 2+ > + + (pcm) Bi 14: Cho 3 s thc dnga, b, c. Chng minh bt ng thc sau: ( ) b a cb aca cbc ba >+++++6491 162 2 2 Du = ca bt ng thc xy ra khi2c b a = =Gii:p dng bt ng thc Cauchy ta c: ( )34942a c bc ba>+++ (1);( )34942b a ca cb>+++ (2) ; ( ) c b ab ac8162> + ++ (3) Cng theo v cc bt ng thc t (1), (2) v (3) ta c: ( ) ( ) c b a c b ab aca cbc ba83498913 162 2 2+ + > + + ++++++ ( ) b a cb aca cbc ba >+++++ 6491 162 2 2 (pcm) 7.K thut Cauchy ngc du Xt bi ton sau:Bi ton:Cho 3 s thc dnga, b, c c tng tha iu kin : 3 = + + c b a . Chng minh bt ng thc sau:

231111112 2 2>+++++ c b a Phn tch v gii:Ta khng th dng trc tip bt ng thc Cauchy vi mu v bt ng thc sau s i chiu:

232121211111112 2 2> + + s+++++ c b a c b a

||||.|

\|=+ +> = > + +233123 12321.21.213212121Do33c b aabcc b a c b a n y chng ta s blng tng trong cch gii. y ta s s dng li bt ng thc Cauchy theo cch khc: http://boxtailieu.netboxtailieu.net 48 (1)212111112222aaaaaa = >+ =+ Tng t ta c:(2)21112bb >+; (3) 21112cc >+ Cng theo v(1), (2), (3) ta c: 23231111112 2 2=+ + >+++++c b ac b a (pcm) Nhn xt:K thut Cauchy ngc du c th hiu l ta lynghch o hai v ca bt ng thc Cauchy sau nhn hai v vi -1. Khi du ca bt ng thc ban u s khng i chiu. Bi 1: Cho 3 s thc dnga, b, c c tng tha iu kin : 3 = + + c b a . Chng minh bt ng thc sau: 23111111>+++++ ca bc ab Gii:Ta c: (1)21211111 abababababab = >+ =+

Tng t ta c: (2)2111 bcbc >+; (3) 2111 caca >+ Cng theo v(1), (2), (3) ta c: ( )23233232 2 2 213213111111= =+ + =|.|

\| +++++ >+ + >+++++c b a a c c b b aca bc abca bc ab (pcm) Bi 2: Cho 3 s thc dnga, b, c c tng tha iu kin : 3 = + + c b a . Chng minh bt ng thc sau: 231 1 12 2 2>+++++ accbba Gii:Ta c: (1)2 2 1 12222ababababababa = >+ =+

http://boxtailieu.netboxtailieu.net 49 Tng t ta c: (2)2 12bcbcb >+; (3) 2 12cacac >+ Cng theo v(1), (2), (3) ta c: ) (1'2 1 1 12 2 2ca bc abc b addcbba + + + + >+++++ Mt khc ta c: ( ) ( )( )) (2'3322 2 2

22 2 2 22 2 2 2 2 2=+ +s + + + + + + = + + =+++++s + +c b aca bc abca bc ab -c b a c b aa c c b b aca bc ab

T (1) v (2) ta c:

232331 1 12 2 2= >+++++ accbba (pcm) Lu:Tassdngktqu ( ) 332=+ +s + +c b aca bc ab trongchngminhcc biton khc. Bi 3: Cho 3 s thc dnga, b, cc tng tha iu kin : 3 = + + c b a . Chngminh bt ng thc sau: 31111112 2 2>++++++++accbba Gii:Ta c:

( ) ( )(1)21211111112222b ababb aabb aaba + + =+ + >++ + =++ Tng t ta c:(2)21112c bcbcb + + >++ ; (3) 21112a cacac + + >++ Cng theo v(1), (2), (3) ta c:

( )323- 323

23- 323 2322311111122 2 2= + =+ ++ >+ + ++ +>+ + + + + + + + >++++++++c b aca bc ab c b aca bc ab c b ac b aaccbba

Vy http://boxtailieu.netboxtailieu.net 50 31111112 2 2>++++++++accbba Bi 4: Cho 3 s thc dnga, b, c . Chng minh bt ng thc sau:

22 232 232 23c b aa ccc bbb aa + +>+++++ Gii: Ta c: (1)2 222 222 23baababab aabab aa = >+ =+ Tng t ta c:(2)22 23cbc bb >+;(3) 22 23aca cc >+ Cng theo v(1), (2), (3) ta c: 2 22 232 232 23c b a c b ac b aa ccc bbb aa + +=+ + + + >+++++

(pcm) Bi 5: Cho 3 s thc dnga, b, c c tng tha iu kin : 3 = + + c b a . Chng minh bt ng thc sau: 21 1 12 2 2>+++++ b aca cbc ba Gii:Ta c:

( ) ( )( ) (1) abc ab ac baac a baac a bac abac bc abac bc abac ba+ >++ > = = >+ =+411 4 2 22 1 122222 Tng t ta c: ( ) (2)4112abc bc ba cb+ >+; ( ) (3) 4112abc ca cb ac+ >+ Cng theo v(1), (2), (3) ta c: 4-434-4 1 1 12 2 2abc ca bc ab abc ca bc abc b ab aca cbc ba + + =+ + + + >+++++ (1)Mt khc ta c: ( )

332ca bc abc b a+ + >+ +=4 43 ca bc ab + +> (2) http://boxtailieu.netboxtailieu.net 51 4 413 33abcabc c b a > > + + =(3) Cng theo v(1), (2), (3) ta c: 341431 1 12 2 2> + ++++++ b aca cbc ba 21 1 12 2 2>+++++b aca cbc ba(pcm) Bi 6: Cho 3 s thc dnga, b, c c tng tha iu kin3 = + + ca bc ab . Chng minh bt ng thc sau: 11 2 1 2 1 23 3 3>+++++ accbba Gii:Ta c: (1)323 121 2233 333abababab bababa = >+ + =+ Tng t ta c: (2)321 23bcbcb >+;(3) 321 23cacac >+ Cng theo v(1), (2), (3) ta c:

( )) (1' 2321 2 1 2 1 23 3 3 + + =+ + + + >+++++c b aca bc abc b aaccbba Mt khc ta c: ( )( ) ) (2' 3 3 3 2= + + > + + + + >+ +ca bc ab c b aca bc abc b a Cng theo v (1) v (2) ta c:11 2 1 2 1 23 3 3+ + + > + + ++++++c b a c b aaccbba

11 2 1 2 1 23 3 3>+++++accbba (pcm) Bi 7: Cho 3 s thc dnga, b, c . Chng minh rng:

32 232 232 23c b aa ca ccc bc bbb ab aa + +>+ +++ +++ + Gii: Ta c: http://boxtailieu.netboxtailieu.net 52 ( )(1)323 32 22 22 23b a b aaabb a abab ab aab b aab ab aa =+ =+ >+ + + =+ + Tng t ta c: (2)322 23c bc bc bb >+ +; (3) 322 23a ca ca cc>+ + Cng theo v(1), (2), (3) ta c:

32 232 232 23c b aa ca ccc bc bbb ab aa + +>+ +++ +++ +(pcm) Bi 8: Cho 3 s thc dnga, b, c c tng tha iu kin3 = + + c b a . Chng minh bt ng thc sau: 12 2 2222222>+++++ a ccc bbb aa Gii: Ta c: ( ) (1)3232 22323 422 2222ab aababab b aabab aa = >+ + =+ Tng t ta c: ( ) (2)3223222bc bc bb >+;( ) (3) 3223222ca ca cc >+ Cng theo v(1), (2), (3) ta c:

( ) ( ) ( )( ) ( ) ( )323322 2 22 2 22 2 2222222(*)3 3 33 3 3((

+ + >((

+ + + + >+++++ca bc abca bc ab c b aa ccc bbb aa Mt khc ta c:( )3. .32b ab ab ab a ab+ +s =3 (1) Tng t: ( )32c bc bbc+ +s3 (2) ; ( )32a ca cca+ +s3(3) Cng theo v (1), (2) v (3) ta c ( ) ( ) ( ) ( ) ( )( )( )333.313 .323.313231322 22 2 2= + =+ ++ + + s+ + + + + s + +c b ac b aca bc ab c b a ca bc ab3 3 3 http://boxtailieu.netboxtailieu.net 53 ( ) ( ) ( ) (**)3 3 32 332322 2 2- . ca bc ab = >((

+ + T (*) v (**) ta c: 1 2 32 2 2222222= >+++++ a ccc bbb aa(pcm) Bi 9: Cho 3 s thc dnga, b, c tha iu kin3 = + + c b a . Chng minh bt ng thc sau: 12 2 2323232>+++++ a ccc bbb aa Gii: Ta c: (1)3232 223 23 633 3332a b aababab b aabab aa = >+ + =+ Tng t ta c: (2)3223 232b c bc bb >+ ; (3)3223 232c a ca cc >+ Cng theo v(1), (2), (3) ta c:

( )( ) 323 322 2 23 2 3 2 3 23 2 3 2 3 2323232(*) c a b c a bc a b c a b c b aa ccc bbb aa+ + >+ + + + >+++++

Mt khc ta c:

3231 2311 . .3 3 2) (1'b ab aba ab a a b a b+=|.|

\| +=|.|

\| + +s = Tng t ta c: ) (2'323 2c bcb c+s ;) (3'32 3 2a cac a+s Cng theo v(1), (2), (3) ta c:

( ) ( )( )(**)33.32332323 2 3 2 3 2=+ +++ +s+ + ++ +s + +c b a c b aca bc abc b ac a b c a b T (*) v (**) ta c: 12 2 2323232>+++++ a ccc bbb aa(pcm) http://boxtailieu.netboxtailieu.net 54 C. MTSKTHUTSDNGBTNGTHC BUNYAKOVSKI I.BT NG THC BUNYAKOVSKICho 2n s thc bt k n nb b b a a a ,..., , , ,..., ,2 1 2 11 , > e n Z n , ta lun c: ( ) ( )( )2 22212 222122 2 1 1... ... ...n n n nb b b a a a b a b a b a + + + + + + s + + +Du = xy ra khi v ch khi nnbababa= = = ...2211(quy c0 =ib th0 =ia ) II.MTSKTHUTSDNGBTNGTHC BUNYAKOVSKI 1.K thut tch ghp b s Bi 1: Cho cc s thc dnga, b, c tha1 = + + c b a . CMR91 1 1> + +c b a Gii: p dng bt ng thc Bunyakovski : ( ) 91.1.1.1 1 1 1 1 12=||.|

\|+ + >|.|

\|+ + + + = + +ccbbaac b ac b ac b a Vy91 1 1> + +c b a Bi 2: Cho cc s thc dnga, b,c. CMR : 6 s+ ++++ ++++ ++c b aa cc b ac bc b ab a Gii: p dng bt ng thc Bunyakovski : ( ) 6 1 1 12 2 22=|.|

\|+ ++++ ++++ +++ + s||.|

\|+ ++++ ++++ ++c b aa cc b ac bc b ab ac b aa cc b ac bc b ab a 6 s+ ++++ ++++ ++c b aa cc b ac bc b ab a http://boxtailieu.netboxtailieu.net 55 Bi 3:Cho cc s thc dnga, b, c tha4 = + + ca bc ab . CMR:3164 4 4> + + c b aGii: p dng bt ng thc Bunyakovski, ta c : ( )( ) ( ) ( )( )( )( ) 16. 1 . 1 . 1 1 1 12 2 2 2 2 222 2 2 4 4 4 2 2 2= + + + + >+ + + + = + + > + + + +ca bc ab ca bc aba c b c b a c b a c b a 3164 4 4> + + c b a (pcm) Bi 4:Cho cc s thc dnga, b. CMRb aabba+ > +Gii: p dng bt ng thc Bunyakovski, ta c : ( ) ( )( )abbab ab aabbaaabbbab a+ s + +||.|

\|+ s||.|

\|+ = + . . .244442 (pcm)Bi 5:Cho cc s thc dng a, b. CMR ||.|

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Bi 10:Cho cc s thc dnga, b, c. CMR( ) ( ) ( ) ( ) c b a b aca cbc ba+ +>+++++ 492 2 2 Gii: Ta c:

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++++++ + 2.K tht chn im ri Bi1: Cho cc s thc dnga, b,c tha6 > + + c b a . Tm gi tr nh nht (GTNN) ca 1 1 1222222accbba A + + + + + =. Phn tch: Chuyn i mt biu thc trong cn thnh mt biu thc ngoi cn. Gi s vi cc s | o,ta c: ( )( )( )( )((

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141112 = = = = === = = = ca bc abaccbbac b a|o| o| o| o , chn ==14|o Kt hp vi k thut chn im ri trong bt ng thc Cauchy ta c li gii: Gii ( )( )( )( )( )( ) ( ) ( ) ( ) | |( )( )( ) ( )( ) ( )( )( ) ( )217 36 29.6 29.813 6 .831171 6 296 2981831171 694171 1 1 194171

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Vi2 = = = c b athGTNN ca A l 217 3 http://boxtailieu.netboxtailieu.net 62 Bi3: Cho cc s thc dnga, b,c tha10 2 > + + + abc c b a . Tm GTNN ca 4 29 8 4 29 8 4 29 82 2 222 2 222 2 22c b acb a cba c baA + + + + + + + + =Do A l biu thc i xng vi a, b, cnn ta d on GTNN ca A t ti 2 = = = c b aS im ri:

141112 = = = = === = = = ca bc abaccbbac b a|o| o| o| o , chn ==14|o Kt hp vi k thut chn im ri trong bt ng thc Cauchy ta c li gii: Gii ( ) ( )( ) ( ) ( ) ( )( )( )6 6247272 2 6 12 6 2 2 2 2 2 2 .42 .42 .42 6 2 2 24 4 4 94 4 4. 24944 29 84 18 2 944 29 84 18 2944 29 84 18 22 2 222 2 222 2 22= > > + + + + >+ + + + + + + + + + >+ + + + + + + + +|.|

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\|+ + > + + s + + + ++ + s + + + ++ + s + + + +Aabc c b ac b a abc abc abc ccbbaac b a ab c ac b bc a ccbbaaca bc ab c b ac b aAca bac b acca bbb a cbca baa c ba Vi2 = = = c b athGTNN ca A l6 6 http://boxtailieu.netboxtailieu.net 63 Ti Liu Tham Kho 1.EE.Vrosovo,NSDenisova,Thchnhgiitonscp,ngidchHongTh Thanh Lim, Nguyn Th Ninh, Nguyn Vn Quyt, NXBGD, 1986. 2.L Duy Thin , S dng bt ng thc Bunyakovski giimt bi ton cc tr i s, Sng kin kinh nghim 2009, Trng THPT Lang Chnh, Thanh Ha. 3.Nguyn Ngc Duy Nguyn Tng V, Bt ng thc Cauchy, Trung tm bi dng kin thc Quang Minh, Thnh ph H Ch Minh. 4.Nguyn Vit Hi, K thut chn im ri trong bt ng thc AM-GM (CAUCHY), Trng THPT chuyn Quang Trung, Bnh Phc. 5.NguynVnMu,BigingChuynngthcvbtngthc,Chngtrnh bi dng chuyn ton, H Ni, 11/12/2009. 6.NguynNgcSang,PhngphpchngminhbtngthcCauchy,Sngkin kinh nghim 2009, Trng THPT Nguyn Hu, Thanh Ha. 7.Phm Kim Hng,Sng to bt ng thc, Nh xut bn Tri thc. 8.Tp ch Ton hc Tui tr. 9.Trn Phng Nguyn c Tn, Sai lm thng gp v sng to khi gii ton, Nh xut bn H Ni, 2004. http://boxtailieu.netboxtailieu.net