ng dng o hm chng minh bt ng thc trong thi i Hc Trn Hong Anh, ngy 19-02-2014 Nick trn din n VMF : Toc Ngan Bi vit tham d Cuc thi Vit bi k nim 10 nm Din n ton hc (2004-2014) Li m u xin gi li cho, li chc n cc thnh vin trong BQT din n, cc Mod, cc thnh vin mnh khe, thnh cng trong cuc sng v mong din n ta s cng ngy cng ln mnh, mnh vit bi ny cng ch c tinh thn giao lu, ng gp cho phng tro ca din n. Mt phn mnh cng ang trong qu trnh n thi H, thi gian gp rt v kh nng bn thn cng c hn nn trong bi vit c iu g hn ch hay sai st mong mi ngi gp cho. Nh mc ch ni trn, bi vit ny chia s 1 s kinh nghim ca mnh khi lm cu 6 trong thi H, tuy cha tht y nhng cng khi qut phn no tng ra v cch gii. Trong bi vit mnh s dng phn ln cc v d c trn din n v thi th ca cc trng chuyn trang web http://nguoithay.vn v 1 s v d trn www.moon.vn m mnh thy ph hp vi tiu ch bi vit Bi vit c chia l cc phn nh sau :Phn 1: Khi qut 3 cu bt ng thc trong thi nm 2013 Phn 2: Cc bi ton 2 bit i xng v khng i xng Phn 3: Cc bi ton 3 bin i xng 2 bin, 3 bin dn v 1 bin ri s dng o hm cc bi ton tng hp, UCT By gi mnh xin bt u vi mt s v d sau :VD1: Cho| |2 2 2, , 1, 3 , 14 a b c a b c e + + = . Tm GTLN ca(1 )(2 )b cPa a= +VD2: Cho, 0 a b >v 2 2 318 16 a b a b + = +Tm GTNN ca6P a bab= + +VD3: Cho, , 0 a b c >v 4 4 43 a b c + + =Chng minh rng : 5 5 5( ) ( ) ( ) 3 ab bc ca + + sVD4: Cho, , 0 a b c > v 2 2 21 a b c + + =Tm GTLN ca 1 1 11 1 1Pab bc ac= + + Hai v d u tin c ly t thi th ca trang web www.onluyentoan.vn v hai bi sau l nhng bt ng thc ht sc quen thuc. Chng ta hy thi bn v kh ca tng bi m cc cn quan tm l hng th ca mi ngi khi bt tay vo gii quyt chng. V phn ln ngi ta s thy hng th hn khi lm VD3 v VD4 hn l hai bi u, n gin l nhng bi ton c pht biu n gin, cc bin i xng nhau, cn VD2 thm ch ta cn khng tm c ngay ng thc xy ra khi no ??? Phn I : Chng ta hy im li 3 cu trong thi 2013, trc ht l khi A VD5: Cho, , 0, a b c >v( )( ) 4 a c b c c + + =Tm GTNN ca 3 3 2 23 332 32( 3 ) ( 3 )a b a bPc b c a c+= + + + Bi lm : Nhn qua biu thc v gi thit thy y l biu thc i xng gia, a b nn c 2 hng i, s dn v theochoc s dng bt ng thc c in, d on lun ng thc khia b c = =Thy biu thc v gi thit ng bc nn nu ta chia c t v mu choc , t ,a bx yc c= =ta s c biu thc 2 bin i xng , khi bi ton c v n gin i kh nhiu. Ta bt tay vo lm T gi thit( 1)( 1) 4 3a bx y xyc c + + = + + =V 3 32 23 332 32( 3) ( 3)x yP x yy x= + ++ + p dng 33 3 3 2 2( )8( )4 3 3u v x yu v P x yy x++ > > + ++ +

Nhn thy, x ynn khi ta t S x yQ xy= + =th3 23( 1) 2 6S QP S S S+ = > + By gi ta ch cn i tm min gi tr caSri kho st hm s choT gi thit, p dng AM-GM ta c 2( )3 24x yx y xy x y S x y+= + + s + + = + >Kho st( ) f Strn |2, ) +ta c c( ) (2) 1 2 1 2 f S f P > = > ng thc xy ra khi0 a b c = = >Cng tng trn ta c th c nhiu bi ton tng t trn nh sau :VD6: Ngi a : Trauvang97 Cho, , 0 a b c > v 2 2 26 4 ( ) a b c c a b + + = +Tm GTNN ca 3 3 2 22 3( ) ( )a b a bPc b a c a b c+= + ++ + Bi lm: tng khng c g khc, min102P a b c = = = >VD7: www.moon.vn Cho| |, , 1, 2 a b c e .Tm GTNN ca 22( )4( )a bPc ab bc ca+=+ + + Bi lm Trc ht ta d on min211 6cPa b= = = = do, a b i xng v mu s c i lngc , khi nu chia c t v mu choc , t,a bx yc c= =2( ) 1, , , 21 4( ) 2x yP x yx y xy+( = e (+ + + Do ta ch cn chng minh 26( ) 1 4( ) x y xy x y + > + + +V ng thc xy ra khi 12x y = =nn ta c th nhm d dng nh sau 2 22( ) 2 4( ) 2( ) 3 4( ) 1 x y x y x y x y + + > + + +> + +Do ch cn chng minh 2 26( ) 8 2( ) 3 x y xy x y + + > + + , lun ng do 1,2x y >VD8: B-2013 Cho, , 0 a b c > . Tm GTLN ca 2 2 24 9( ) ( 2 )( 2 )4Pa b a c b ca b c= + + ++ + + Bi lm : S dng AM-GM ta c c 224 92( )( )433Pa b ca b cs + ++ ++ n y kho st 2( )( ),3a b cf t t+ +=ta c pcm Nhng bi ton kiu th ta s xt phn sau VD9: D-2013 Cho, 0 x y > tha mn1 xy y s Tm GTLN ca 2 226( )3x y x yPx yx xy y+ = + + Bi lm Nhn thyPl biu thc ng bc v khi nhnPta cng kh c th dung bt ng thc g nh gi c, cng kh d on lun ng thc xy raChia c t v mu choy , t 21 26( 1)3x t tt Py tt t+ = = + + By gi s dng gi thit tm min gi tr catri kho st hm s p dng AM-GM ta c 21 1 1 11 (0,4 4xxy y ty y y(s s s e( ng thc xy ra khi2 y =n y bi ton dn c gii quyt, cng vic ch l kho st hm s( ) f ttrn 1(0,4(( Mt s v d tng t sau :VD10: Cho, 0 x y > v 2 2 4 42 (11 1) 8 6 1 y x x y + = + +TmGTNN ca 22 2 2 2( )( 4 )x yPx y x y y=+ + + Bi lm tng ng bc, s dng gi thit tm min gi tr ca xty=T gi thit ta c 2 2 4 4 4 2 22 42 1 1 52 (11 1) 8 6 1 8 22 6 14 2y x x y t t ty y+ = + + + = s s sXt 22 2( )( 1)( 4 1 1)tP f tt t= =+ + +, t 221 5,4 2( ) ( )( 1)( 4 1 1)at aaf t f aa a(e( = = =+ + + Kho st ta c c 2 1( ) ( )5f t f a= >ng thc xy ra khi 121xy== VD11: Ngun : http://nguoithay.vn Cho, 0 a b >v 2 23 1 (3 2) a b b a + + = +Tm GTNN ca 2 222 22 2 823a b a ab bPab ba b+ + += ++ VD12: Ngun VMF Cho, 0 a b >v1 a ab > +Tm GTLN ca 2 23abPa b=+ Nh thy c 3 bi thi H nm va ri u c th x l theo 1 phng php chung, nh gi a v 1 bin, tm min gi tr ri kho st hm s tm cc tr, v y cng l xu hng ra thi th ca cc trng, cng nh thi tht hng nm, ngoi vic dung cc bt ng thc AM-GM, Cauchy-Schwarzt nh gi cc bin, chng ta cng cn s dng 1 cng c rt hu hiu trong chng trnh THPT, chnh l o hm, v y cng l con ng chnh trong li gii cc v d trong bi vit. Phn II : Cc bi ton 2 bin i xng v khng i xng cc bi ton 2 bin i xng, ta c phng php dngnhiu l a v dng S x yP xy= + =, do tnh i xng nn ta lun lun a biu thc v dng biu din nh th n ngay vi khi D-2012 VD12: Cho, 0 x y >v 2 2( 4) ( 4) 2 32 x y xy + + sTm GTNN ca 3 33( 1)( 1) A x y xy x y = + + + Bi lm T gi thit 2( ) 8( ) 0 8 x y x y x y + + s + sNhn thyA khng th t GTNN ti tm4 x y = = , ta vit li nh sau

3( ) 3( ) 3 3 A x y x y xy = + + +p dng AM-GM ta c 2 233( ) 3( )3 ( ) 3( ) 34 4x y x yxy A x y x y+ +s > + + +n y mi chuyn r, t|233(0,8 ( ) 3 34tt x y t A f t t t = + e > = +ng thc xy ra khi... x y = =VD13: Chuyn L Qu n ln 2-2013 Cho, 0 x y >tha mn 3 1 xy x y = + + .TmGTLN ca 2 23 3 1 1( 1) ( 1)x yAy x x y x y= + + + Bi lm Ta d on ng thc xy ra khi1 x y = =Khi gp nhng bi ton ny, khi c xong gi thit ta suy ra lun iu kin ca, S P Ta c3 1 2 1 1 xy x y xy xy = + + > + > , hin nhin2 x y + >By gi ta s a biu thc v dng hm s cha binP ( hocS ), s dng gi thit, quy ng ta c 25 1( )4PA f PP= =T ta c c( ) 1 A f P = sng thc xy ra khi1 x y = =VD14: Cho, 0 x y >tha mn 4 464 x yxy = + +Tm GTNN ca 2 21 1 3 21 2 1 2 5xyAx y x y= + ++ + Bi lm i xng 2 bin, t gi thit ta c 4 4 2 264 2 4 1 x y x y xy Pxy = + + > + = sD on lun min1 1 P x y = = = n y ta c th quy ng , a v dng tng tch, t gi thit ta c

4 4 4 2 2 2 26 64 ( ) 6 4 ( ) 4 x y x y x y xy x yxy xy+ + = + + + = a c dng tng tch nhng nu thStheoP hay ngc li ta s c 1 biu thcrt cng knh, v th ta phi s dng cc bt ng thc ph lm n gin biu thc cho. Ta c bt ng thc quen thuc sau

2 21 1 21 1 1 UV U V+ >+ + + vi1 UV >Hoc2 21 1 21 1 1 UV U V+ s+ + + vi1 UV sKhi ta c 2 trng hp sau :TH1: 1,14P xy (= e ( , khi 2 .2 1 x y >p dng bt ng thc trn 1 1 2 21 2 1 2 31 2x yxy + > >+ ++ V 22 23 2 1( ) 4(1 ) 03 5xyx y xyx y> + > 2 113 3P > + =TH2: 1 1 1(0, ) 1 14 1 2 1 2P xy Px y= e + > >+ + Kthp li ta c min1 1 P x y = = =Tr li vi bt ng thc ph trn, l pht biu kh n gin nhng rt hay v quan trng trong nhng bi i xng 2 bin, thm ch 3 bin c dng phn thc nh trn. Ta c 1 s v d p dng sau : VD15: Cho| |, , 1, 9 , , a b c a b a c e > >Tm GTNN ca 2a b cPa b b c c a= + ++ + + VD16: H Vinh ln 2-2013 Cho, 0 x y >tha mn 4 423 3 xy x yxy+ = + +Tm GTLN ca 2 22 2162P x yx y= ++ + VD17: Cho, 0 x y >tha mn 4 412 x y xyxy+ + = +Tm GTNN ca 2 22 2 31 2 1 1Pxy x y= + + + + Xt n dng ton khc sau :VD18: Ngi ng: Ispectorgadet Cho, x y ev2 2 1 1 x y x y + = + + +Tm GTNN v GTLN ca 2(1 )( ) ( )2 2xy xy x yP x y y xx y+= + ++ Bi lm Vit 2( ) 22x yPx y+= ++ By gi ch cn tm min gi tr cat x y = +ri kho st 22( )2tf tt= +S dng Cauchy-Schwarzt ta c

2 21 2 2 1 (2 1 )( 2 1) 1 6 x y x y x y t x y + = + + s + + + s = + sKho st hm s ta c minmax5( , ) (2, 1)2218 ( , ) (6, 0)6P x yP x y= = = + = VD19: Cho, a bev1 2 4 1 a b a b + = + +Tm GTLN v GTNN ca 21( ) 9 P a b a ba b= + ++ Bi lm Ta cn tm min gi tr cat a b = +p dng Cauchy-Schwarzt ta c1 2 4 1 2 2 1 3( 1) 1 4 a b a b a b a b t a b + = + += + + s + s = + sKho st hm s ................ VD20: Onluyentoan ln 1-2012 Cho 2 2, 0, 2 x y x y > + =Chng minh rng 3 22942x yPx y y= + >+ Bi lm Cch 1: Theo minhtuyb p dng Cauchy-Schwarzt ta c| |22( 2 ) ( 3 )xP x x y yy+ + > +Li c 2 22 13 3 2( ) 4x yy y yy y y+ = + = + > 2 2( 2 ) 2( ) 2 2( ) 4 x x y x y x y + + = + s + =Vy ta c pcm, ng thc xy ra khi1 x y = =Cch 2: Bt ng thc tng ng 3 4 2( 2 ) 9 4 ( 2 ) x x y y y x y + + > +Ta s to ra bt ng thc di dng ng bc, hay vit li thnh

3 4 2 2 2( 2 ) 9 2 ( 2 ) 2( ) x x y y y x y x y + + > + +t 2 6 5 4 3 2( 1) ( 6 14 12 43 66 49)xt t t t t t t ty= + + + + + + > Vy ta cng c pcm R rng cch s 2 khng hay v t nhin bng cch s 1 Ta c 1 s v d p dng sau:VD21:H Vinh ln 4-2013 Cho 2, 0, 2 12 a b a b > + = . Tm GTNN ca 4 4 44 4 58( )Pa b a b= + + VD22: Cho, 0, 2 a b a b ab > + =Tm GTNN ca 2 28 4 1a bPb a= ++ + VD23: Ngi ng quanghao98 Cho, , 2 4 1 2 2 8 x y x y x y e + = + + +TmGTLN v GTNN ca2 16 P x y = + Kt thc phn 2, ta n vi phn 3, phn quan trng nht, trc ht xt 1 s v d sauVD24: Cho 2 2 2, , 0, 1 a b c a b c > + + =Tm GTLN ca(1 2 )(1 ) P a bc = + +VD25: Cho| |, , 1, 4 , 2 8 a b c a b c e + + =Tm GTLN ca 3 3 35 P a b c = + +Bi lm: Nhn 2 v d trn, ta rt ngay c nhn xt: c 2 bin c vai tr nh nhau ( v d on cc tr t c khi 2 bin bng nhau ) trong gi thit v biu thc, ng nh tiu ch ca phng php o hm, ta s tm cch a biu thc v dng hm s ca bin cn li ri tin hnh kho st Vi v d 24, ch cn p dng AM-GM ta c

2 2 21(1 2 )(1 ) (1 2 )(1 ) (1 2 )(1 ) ( )4 4b c aP a bc a a f a+ = + + s + + = + + =Kho st vi(0,1) aeta c 26 2 6 37 1'( ) 02 6a af a a + = = =Lp bng bin thin ta c 37 1( ) ( ) ...6P f a fs s =ng thc xy ra khi 2 2 237 161ab ca b c==+ + = v d 25, d onmax137 ( , , ) (1,1, 3) P a b c = = , v ta s a v dng( ) P f c sX l i lng 3 3a b +nh sau 3 3 3( ) 3 ( ) 2 (8 2 ) 3 (8 2 ) 5 P a b ab a b c c ab c c = + + + = +n y ta khng th nh gi 2 2( ) (8 2 )4 4a b cab+ s =v khi 23 33(8 2 )(8 2 ) (8 2 ) 54 cP c c c> + , khng th tm c GTLN ( 1)( 1) 0 1 3 3( 1) a b ab a b ab a b > > + s +3 3(8 2 ) 3(7 2 )(8 2 ) 5 ( ) P c c c c f c s + =T gi thit ta c| |1, 3 c e , kho st ta c( ) (3) 137 137 f c f P s = sng thc xy ra khi1, 3 a b c = = =VD26: Ngi ng phanquockhanh Cho, , 0 a b c >v1 abc = .Tm GTNN ca 1 1 22 1 2 1(2 1) 6 3Pa bc c= + ++ ++ + Bi lm Theo daicahuyvn NhnPi xng, a b nn ta s vit di dng( ) P f c >Thy i lng quen thuc 2 21 11 1 U V++ + nn nu 1 1 1 2 22 .2 14 2 1 2 1 2 12 1 2c cab a ba b cab c> > + > = >+ + ++ + Khi 22 1(2 1) 6 3cPcc c> +++ +, 28( 1) (6 33) 09P c c > s , lun ng By gi ta cn xt nt trng hp 14ab sKhi d thy 1 1 812 1 2 1 9 a b+ >>+ + Vy min819P a b c = = = =Trong nhng bi ton 3 bin c 2 bin bng nhau, cng vic l dn v bin cn li, v khi vic s dng cc bt ng thc ph nh gi l 1 vic rt quan trng, gip ta dn bin nhanh hn v hiu qu hn. Chng ta n vi v d tip sau VD27: Ngi ng : Trannhuphuc Cho 2, , 0, 1 1 2 1 2 5 x y z x y z > + + + + + =Tm GTLN ca 3 3 32 P x y z = + +Bi lm Ta s dung bt ng thc ph sau1 1 1 1 a b a b + + + > + + +ng thc xy ra khi0 ab =Khi 1 2 1 2 1 1 2 2 y z y z + + + > + + + , ng thc xy ra khi0 yz =2 2 25 1 1 2 1 2 1 1 2 2 1 2 1 2 2 x y z y z x x y z = + + + + + > + + + + + > + + + +22( ) 8 x y z + + s trn ta s dng iu kin0 yz =nn khi 3 3 3( ) y z y z + = +Xt 23 3 3 3 3 3 382 2 ( ) 2 ( ) ( )2xP x y z x y z x fx= + + s + + s + =Kho st hm s vi0, 2 2 ( ) (0) 64 64 x fx f P (e s = s ng thc xy ra khi( , , ) (0, 4, 0) (0, 0, 4) x y z = =R rng v d 27 kh hn 2 v d u v phi s dng bt ng thc ph kh hay v d on lun c ng thc xy ra khi no ? VD28: Chuyn Vnh Phc ln 1-2013 Cho 2 2 2, , , 3 x y z x y z e + + =Tm GTLN ca 2 23 7 5 5 3 7 P x y y z x z = + + + + +Bi lm Cch 1: Trc ht ta vn s tm cch a v dng( ) P f x sp dng AM-GM ta c ngay 2 2 2 2 2 26 12 2( ) 6 12( ) 6 12 6 23 3 3 3 103 3 3x y z x y z x xP+ + + + + s s = sCch 2: Vit( ) P f t svit y z = +p dng AM-GM ta c 22 26 7( )5( ) 2 5( ) 14( ) 2 18 6( )2x y zP y z y z y z y z+ + ( s + + = + + + + + Li c 22 2( )2y zy z++ >nn vit theotta c2 2 2 2 2 2( ) 5 6 14 36, 0 2( ) 2( ) 6 P f t t t t t y z y z x y z s = + + + s = + s + s + + =25 7 6'( ) 0 226 14 36tf t ttt t = + = = + + Lp bng bin thin( ) (2) 3 10 3 10 f t f P s = sng thc xy ra khi( , , ) (1,1,1) ( 1,1,1) x y z = = VD29: Ngun VMF Cho 3 3, , 0, ( 1) a b c a b c c > + = Tm GTNN v GTLN ca 2 2 22( )a b cPa b c+ +=+ + Mt s bi ton i xng 3 bin s dng o hm VD30: Ton hc tui tr Cho 1, , , 33a b c (e ( Tm GTNN v GTLN ca a b cPa b b c c a= + ++ + + Bi lm Trc ht d on min7 1( , , ) ( ,1, 3)5 3P a b c = =v hon vVit 1 1 11 1 1Pb c aa b c= + ++ + +, c i lng 2 21 11 1 U V++ + nn nu. 1b cc aa b >>Th 1 1 2 1 21 1 11 1Pb c ac ca b ca a+ > > ++ + ++ +, khi ( ),cP f t ta> =Ta bt u lm, do vai tr ca, , a b cnh nhau nn ta c th gi smax c =| |2222 1 2, 1, 311 1 11t cP tt t a tt > + = + = e+ + ++ Kho st ta c 7( ) (3)5P f t f > > =ng thc xy ra khi 1, , , 3313 ( , , ) ( ,1, 3)3a b cct a b cab ca b(e( = = == v hon v Cng vic tm GTLN hon ton tng t khi gi smin c =1 1 21 11b cca ba + s+ ++, do. 1b c ca b a = sVD31: Cho, , 0 a b c >v1 a b c ++ =Chng minh rng 4 4 41( ) ( ) ( )12a b c b c a c a b + + sBi lm Nhng bi ton kiu ny thng c ng thc khi 1 bin bng 0 Ta lm nh sau: Do vai tr ca, , a b cnh nhau nn ta c th gi s0 a b c > > >4 4 4 4 4( ) ( ) ( ) P a b c ba ca a b c a b c s + + + = + + +t1 t b c a t = ++ =v 4 4 3 3( ) (1 3 ) P at a t at a b at at s + = + = t 2( ) 10, , ( ) (1 3 )4 4a tx at x P fx x x+(= s e s = ( Ta c lun 1 1( ) ( )6 12P fx f s s =ng thc xy ra khi 3 3 3 3( , , ) ( , , 0)6 6a b c+ =VD32: Chuyn Vnh Phc ln 5 Cho, , 0, 3 a b c a b c > ++ =Tm GTLN ca 2 2 2 2 2 2( )( )( ) P a ab b b bc c c ca a = + + + Bi lm Do iu kin bi ton nn ta d on max12 ( , , ) (2,1, 0) P a b c = =Gi s0 a b c > > > , suy ra2 2 2 2 2 2 2 2 2 2( ) ( ) 3 ( ) 3 ( ) (9 3 )( ) P a ab b a b a b ab a b a b c ab ab ab ab(( s + = + s + + = t 2 2( ) ( ) 9 90,4 4 4 4a b a b ct ab t+ + +(= s s = e ( Kho st hm s( ) (2) 12 12 f t f P s = sVy ta c pcm VD33: Ton hc tui tr Cho 2 2 2, , 0, 3 a b c a b c > + + =Tm GTNN ca 2 2 2 2 2 216 11ab bc caPa b ca b b c c a+ + += ++ ++ + + Gi : Chng minh 2 2 2 2 2 2a b c a b b c c a ++ > + +Mt s bi tp khc VD34: Cho| |, , 2, 4 a b c eTm GTLN ca 2 2 2 2 2 21( )a b b c c aPabc a b c+ + +=+ + VD35: Ton hc tui tr Cho, , , 1 a b c a b c e ++ =Chng minh rng 3 3( )( )( )18 18a b b c c as sMt s bi tp tng hp VD36: Ton hc tui tr + Chuyn H Vinh Cho 2 2 2, , 0, 3 x y z x y z y > + + sTm GTNN ca 2 2 21 4 8(1 ) ( 2) ( 3)Px y z= + ++ + + Bi lm Cch 1: p dng bt ng thc ph sau 2 2 21 1 8( ) a b a b+ >+ Khi 2 2 221 4 1 1 8(1 ) ( 2) (1 )( 1) ( 2)2 2y yx y xx+ = + >+ + ++ + + 22 28 8 64( 3)( 2) ( 5)2 2Py yzx x z > + >++ + + + + By gi ta cn tm 2yx z A + + sm vn m bo 2 2 21 1212 32 23yxx zyx zyx y z y+ = += = + + = + =+ + = Xt 2 2 2 22 2 2 1 1 2 ( 2) 6 6 A x y z x y z y y y y = + + s + + + + s + + = + s2641(3 5)P > >+, ng thc xy ra khi 12x zy= = = Ta cng tham kho cch dn v binyrt hay sau Cch 2: Theo WhjteShadow p dng AM-GM v Cacuchy-Schwarzt ta c 2 2 21 8 27(1 ) ( 3) ( 4) x z x z+ >+ + + + T gi thit 22 2 2 2( )32x zy x y z y+> + + > +2 22 2( ) ( 4)3 4 42 6x z x zy y y+ + + + > + + > +2 2 2 2 21 8 9 4 9( )(1 ) ( 3) 2(3 4 ) ( 2) 2(3 4 )P fyx z y y y y y + > > = ++ + + + + Xt | |3 22( 2)(2 181 400 236)'( )2 ( 4)( 1)( 2)y y y yf yy y y + + += + + ( ) (2) 1 P fy f > > =Cch 3: Theo nguyenthehoan S dng Cauchy-Schwarzt ta c 2 2 2 2(1 ) 2(1 ), ( 3) 4( 3) x x z z + s + + s +2 2 2 2 2 2 2 2 21 8 1 2 1 1 1 9(1 ) ( 3) 2 2 3 2 2 3 3 2( ) 8 x z x z x z z x z + > + = + + >+ + + + + + + + + 2 24 9( 2) 2(3 ) 8Py y y > ++ + Ta c 2 21 ( 2) (2 10 9) 0 P y y y > + + >Vy c 3 cch lm u cho ta p s ng !!! VD37: http://nguoithay.vn Cho, , 0 a b c >Tm GTNN ca 1 16 4Pa ab bc a b c= + + + + Bi lm Nhn biu thc ta d on ngay( ), P tt a b c > = + +Do ta phi vit c6 4 ( ) a ab bc x a b c + + s + + , do cc hng t ng bc p dng AM-GM ta c6 2 3 .3 3 3 ab a b a b = s + 4 2 .4 4 bc b c b c = s +Cng 2 bt ng thc trn li ta c pcm 21 1 1( 1) 1 14( )2Pa b ca b c a b c > = > + ++ + + + ng thc xy ra khi 3 31 1 14 ( , , ) ( , , )9 9 3612a bb c a b ct a b c== = = + + = VD38: http://nguoithay.vn Cho, , 0, 1 x y z x y z > + + =Tm GTNN ca 3 3 314( 1) ( 1)( 1)x y zPx yz y xz z xyz x y= + + ++ + ++ + + Bi lm D thy vai tr, x yl nh nhau nn ta s vit( ) P z >T gi thit, 2 2( 2) ( 1)1 ( 1)( 1)4 4x y zx y z x y z xy+ + ++ + = + + = + s =3 3 32141( 1)( 1).24x y zPzx yz y zx zz > + + +++ + ++ 3 3 32 24 28( 1) ( 1)x y zPx yz y zx z z > + + ++ + + + By gi cn nh gi 3 3x yx yz y zx++ + theozp dng Cauchy-Schwarzt ta c

3 3 4 4 2 2 2 2 22 2 2 2( )1 2x y x y x y x yx yz y zx z x xyz y xzy x y xyz+ ++ = + > >+ + + + + + + M 2 2 2 2( ) ( 1)1 2(1 ) 2(1 )x y x y zz z z+ + > =+ + + nn 2 3 3 22 2 2( 1) 4 28 9 57( )2(1 ) ( 1) ( 1) 2( 1)z z z z zP fzz z z z +> + + = =+ + + + t 23451(3 5)(3 37 23)3( ) 2 ( ) '( )( 1)zz z zg z fz gzz + + += =+ 5 53 53( ) ( )3 8 8fz f P > = >ng thc xy ra khi 1 1 5( , , ) ( , , )3 3 3x y z =VD39: Chuyn L Hng Phong Cho 2 2 2, , 0, 2( ) 2 x y z x y z x y z xy > + + s + + Tm GTNN ca 2 240 4021 3P x y zy z z= + + + ++ + + Bi lm: Ta hi vng1 3 2 y z x y z x ++ = + + = +v1, 3 y z x + + +u l s chnh phng Pnhn gi tr hu t S dng 1 1 4, 22a ba ba b a b++ > + s+ ta c40 40 160 160 80 21 3 1 3 1 3 422y z x y z x y z x x y z+ > > =+ + + + + + + + + + + + + + n y ta c tng a v bint x y z = + +Ch gi thit 22 2( )2( ) ( ) 42x y zx y z x y z x y z+ ++ + > + + > + + sKhi d on du bng 2 14 3x y z xx y z y z+ = + = + + = + = Gii quyt 2 22 x y z + +nh sau 2 2 2 22 1 1 2 2 2( ) 2 x y z x y z x y z + + = + + + + > + + 80 2 80 22( ) 2 2( 4) 104 4P x y z x y zx y z x y z > + + + = + + + ++ + + + + + t4 (2, 2 2 t x y z t(= + + + e Kho st hm s( ) (2 2) 46 f t f > =ng thc xy ra khi1, 2 x y z = = =VD40: Ton hc tui tr Cho 2 2 2, , , 1 x y z x y z e + + =Tm GTNN ca 228( 2 )( ) 2P xy yz xzx y z xy yz= + + + + + Bi lm Vit li 28( 2 )3 2P xy yz xzxy yz xz= + + + + + t 282 ( )3xy yz xz t P f t tt+ + = = = + Ta cn tm min gi tr catDo 2 2 2 2 2 2( ) ( ) 0 2( ) 2( 2 ) 0 1 x y z x z y x y z xy yz xz t + + + + + > + + + + + > > V 222 3(2 3) 2 .2 2yx xy + >

222 3(2 3) 2 .2 2yz yz + > 2 2( 3 1) ( 3 1) 2 .( 3 1) x z xz + > Cng cc bt ng thc trn li ta c 14 2 3t s Kho st hm s ta c( ) ( 1) 3 P f t f = > = ng thc xy ra khi 2 2 201 1( , , ) ( , 0, )12 2x y z y x zx y zx y z+ + = = + = =+ + = Nhn xt: Vic tm max cat trn l khng cn thit, nhng cho y v chc chn ta nn tm c chn trn v chn di ca bin VD41: www.moon.vn Cho, , 0, 1 a b c a b c > ++ =Tm GTNN ca 21 1 13 9 6 36 2(2 1)Pb c a= + ++ + + Bi lm tng a v binatheo bt ng thc sau 21 1 4 1 (2 1) 13 9 6 36 12 36 6 36 12 36 6 36 6 4 b c b c b c a++ = + > =+ + + + + + + ng thc xy ra khi 2 1212 36 6 36b cb c= =+ + Khi 21 1( )6 4 2(2 1)P f aa a> + = + Xt 1 1 3 3'( ) 0 ( ) ( )2 2 8 8f a a f a f P = = > = >ng thc xy ra khi 1 1 1( , , ) ( , , )2 3 6a b c = VD42: chuyn Lo Cai Cho, , 0 x y z > tha mn1 x y z + + =Tm GTLN ca 3 32( )( )( )x yPx yz y xz z xy=+ + + Bi lm Bt ng thc i xng, x ynn 2 3 3max( )( )( ) P x yz y xz z xy kx y + + + >S dng gi thit ta c 3 32 3 3( ) ( 1) ( 1)x yPx y x y=+ + + S dng AM-GM ta c 3 3 2 23 3 3 34 .( 1) ( 1) 4( 1) ( 1)x y x yPxy x y x ys =+ + + + Ta c 2 233271 1 3 ( 1)2 2 4 4x x x xx x + = + + > + >Tng t ta c c 2327( 1)4yy + >min4( , , ) (2, 2, 5)729P x y z = =VD43: http://nguoithay.vn Cho, , 0, 2 x y z x y z > + + =TmGTLN ca 3 32(2 )(2 )(2 )x yPx yz y xz z xy=+ + + VD44: http://nguoithay.vn Cho, , 0 a b c >tha mn 2 2 23 2( ) b c a a bc + = +Tm GTNN ca 22 22 2 5 4 4( ) ( )a aPbc a b a c += + ++ + Bi lm Ta s vit( ) P f a >T gi thit ta c 2 2 22 2 3 1 1 a a bc b c bc = s sBy gi s l i lng 2 21 1( ) ( ) a b a c++ +theo, bc aNu 2 2 2 22 21 1 1 1 1 2( ) ( ) ( ) ( )( ) ( )b ca b a c a b a ca bc a bc= = = + =+ + + ++ +, ta hi vng bt ng thc ng, nhng rt tic li l 1 bt ng thc sai Vit 2 221 1 1.( )(1 )ba b aa=++ v 2 221 1 1.( )(1 )ca c aa=++ 2 2 22 21 1 1 1 1( ) ( )(1 ) (1 )b ca b a c aa a ( ( + = + (+ + (+ + ( S dng bt ng thc ph 2 21 1 1, , 01 (1 ) (1 )x yxy x y + > >+ + + 2 2 2 2 22 221 1 1 1 1 1 1 1.( ) ( )(1 ) (1 ) 1b c bca b a c a a a bca a a ( ( + = + > = (+ + + (+ + + ( Vy 222 22 2 5 4 42 2 51a aP a abc a bc a +> + > + ++ +, do1 bc sn y ta c th kho st hm s hoc lm nh sau : vit c nh trn ta gin tip s dng iu kin1 a bc = = , nn nu cch lm ng th( ) (1) 7 f a f > =Ta s chng minh 2 222 24 ( 1) ( 1)2 2 5 7 01 1a a aa aa a + + + + > >+ + Vy ta c pcm ng thc xy ra khi1 a b c = = = VD45: http://nguoithay.vn Cho, , 0 a b c >tha mn 2 2 2 2( ) ( ) 8 a b a c b c + + + + + =Tm GTLN ca 2 2( ) ( ) 1 a b c b a cPa c b c c+ += + + + Bi lm: tng dn v kho st( ) P f c sp dng bt ng thc sau 2 2 2( ) x y x yu v u v+s ++ 2 2 22( ) ( ) b c b a b cc b aca c a a c+ + s + s ++ + Tng t ta c 22( ) a b cb aca c+s ++ 2 2 21 1( ) 4 ( ) P a b c a b c f cc c s + + + = = , do gi thit n y c th kho st hoc p dng lun AM-GM ta c

2 2331 1 1 1 33 42 2 44c c Pc c c+ = + + > s ng thc xy ra khi VD46: Cho, , 0, 1 a b c a b c > ++ =Tm GTNN ca 1 1( 1)( 3)( )( ) ( )( )P c a ba b b c c a a b= + + + + ++ + + + Bi lm Bi ny kh n gin nh sau: S dng AM-GM ta c 221 1 1 1 4 4( ) ( 1)( 3) . ( 1)(4 ) 3 4 ( )( ) 1 1P c a b c c c c f ca b a c b c c a c b c c= + + + + + > + + = + + =+ + + ++ + Kho st ta c( ) (0) 8 8 f c f P > = >ng thc xy ra khi 1 1( , , ) ( , , 0)2 2a b c = VD47: Cho, , 0 a b c >tha mn 101 1 11a b ca b c+ + = + + = Tm GTNN v GTLN ca 2 2 2P a b c = + +Bi lm 2 2 2 2( ) 2( ) 100 2 P a b c a b c ab bc ac abc = + + = + + + + = Do ta ch cn tm GTNN v GTLN caQ abc =Ta c 21 10 1 (10 )1 ( )1b c a a a aQ abc f abc a bc a a+ = = = = = Lp bng bin thin ca hm s ta khng tm c cc tr, do hm s gin on ti 1, 8 a a = =VyPkhng c GTNN, GTLN Bi tp trn ging B-2010 Cho, , x y z ev 2 2 201x y zx y z+ + = + + = Tm GTLN ca 5 5 5P a b c = + +VD48: Cho, , a b c ev 2 2 201a b ca b c+ + = + + = Tm GTLN ca 2( ) P abc =Bi lm:Vi nhng bi ton i xng nh th, ta lun vit c di dng hm s no Ta c 2 2 22 2 2( ) 10 ( )2 2a b ca b c a b c bc a ++ + = = + = = n y ta cn xt 2 trng hp TH1: 2 2 2 21 10 ( )2 2bc a P a a s s = t 2 21 10, , ( )2 2t a t P t t (= e = ( Kho st hm s ta c 1 1( ) ( )6 54P f t f = s =ng thc xy ra khi 22 2 21601t abca b ca b c= =s+ + =+ + = TH2: Tng t ta cng c 2 1( ) ( )3 54P f t = s =Kt thc bi vit s l phng php UCT ( thng bit n di dng phng trnh tip tuyn ), dung cho nhng bi m cc bin c lp vi nhau, c vit di dng hm s( ) ( ) ( ) P f a f b f c = + +Ti liu v vn ny c rt nhiu trn din n, c th tham kho ti liu sau:Phng php h s bt nh http://diendantoanhoc.net/forum/index.php?/topic/90839-ph%C6%B0%C6%A1ng-ph%C3%A1p-h%E1%BB%87-s%E1%BB%91-b%E1%BA%A5t-%C4%91%E1%BB%8Bnh-uct/ Phng php UCT V Quc B Cn, Nguyn Thc V Hong http://diendantoanhoc.net/forum/index.php?/topic/76805-ph%C6%B0%C6%A1ng-phap-utc-vo-qu%E1%BB%91c-ba-c%E1%BA%A9n-nguy%E1%BB%85n-thuc-vu-hoang/ a ch: Trng THPT chuyn KHTN S in thoi : 0986504770 Trang web hc tp:[1] http://diendantoanhoc.net/forum/ [2] http://www.artofproblemsolving.com/Forum/index.php