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  <<      -  H   G   =    ?       e(   =  {  #4       e(  −+=  {   *  #%   {{  +=+  4 ( ) ( )( ) i{{  >−−=−−+=−   {{  >⇒    {{  <⇒  -H  G {{{  = O)VcP  ;,- {G t Vc'  K(;!a(;/X(/YO)Vc3P   (;!a*#%c(/ −  +(;/X(/?(Vc G P  #4/*<H    JK-F,-K(;/X(HVc0B+  G <⇒  ⇒ /! \-*_67 "++/!4[#$$=)_   ≥++  [7 JK ( )         −= − + ⇔−=+    =++⇒ ()AAB*D ZEO^  ( ) ( )  |    ≥++⇒ ++≥++⇒ ++=≥++    915=- ⇔==⇔  ∆/>, \-*_87 Cho a,b,c lµ c¸c sè d¬ng. Chøng minh r»ng: =:>#?>?'>@A*%B%C?6D'( ` ++ + + + + + (1.1) [7 Ta sẽ sử dụng BĐT Cauchy nh sau: Ta có a 3 +b 3 a 2 b + ab 2 a 3 +b 3 ab(a+b) + + ; Tơng tự, ta cũng có: A + ++ + . Cộng theo từng vế các BĐT trên lại với nhau ta đợc BĐT cần chứng minh. \-*_9^Cho a, b, c là các số dơng. Chứng minh rằng: ++ + ++ + ++ [^ Ta có: a 3 +b 3 a 2 b + ab 2 a 3 +b 3 + abc a 2 b + ab 2 +abc a 3 +b 3 + abc ab(a+b+c) A popo ++ = ++ ++ Tơng tự , ta có: } po A po ++ ++++ ++ Cộng các BĐT này lại với nhau theo từng vế ta đợc BĐT cần chứng minh. 1.1.1.1. ab$)c"dEE <E"&$1$+>( Z` !a&$ 1$(>b$:/b !c!da/*$, '&$1$$9-I$F'(Z5&$1$9E+5 '(Z!/>b$eV%$0$E\$a&$1$ =:>#?>?'>@A*%B%C?6D'( e  $EP!&$1$Cauchy /!#$$E"f&$1$4 J5$&b(g$:$!$,!"$Y !"#$% h_ \-*_67" ≥ M<\$E.e&$[5$S=+   ,-#&4%[ y~E=(;()&A   #4R/YB/(R   w r ] | j s i   d i     w  r  ]  |  j  s  i      d i  R    w w  r r  ] ]  | |  j j  s s  i i        d i i  K @(=(;()2'(0KR4 U#4V/OB•/; #(NB/'(€KR%(-_C29YBW(Y,#4@2 *<MO(-P(R€  i /@@(`9(Y-D(^€b •2/15$(‚=-B2,P@(/(>,'(Nc(!=( P,A)@(`9(Y-D(^€b'$YgBa2/15$( -(;!c#4   #K ≠   l"4M(=/_gBa2/1 5$(7!c          A α @(`/(Y-D(^€bK    = α 5 4O *</S`/(Y-D(b,/:^ RD/S^ αα  =  ⇒  ⇒ α    =  ⇒    =  V/O(;/X(ZD/S`9(Y-D(b/*<),-) y[7Tfg   f    +       s g s j ≥     s ⋅ g s j ⋅ f  i J%#U(€K(R€  i \-*_87" ≥ A<\$E.e&$[5$@ij    yRD/S/(Y-D(^ αα  =  =:>#?>?'>@A*%B%C?6D'( F € s = α  αα  =  ⇒ ⇒  α  w  =  ⇒ w   =  y[^R€ƒ    €    ++      jj   ƒ j ] ≥     jj   ⋅⋅ ƒ j ] ⋅ € w s JU(€K(R€ w s \-*_9^" ≥ k<\$E.e&$[5$@i  j  j yRD/S/(Y-D(^ αα ]  =   ⇒ ⇒ α ] ] j =  ⇒ ] jj =  y[^R€  ƒ  j €    +      j ]  ƒ         − ]      ≥    j ]  ⋅ ƒ         − ]     €] ]  ƒ  ]            − ≥ ]  ] ]   ] ]]         −+ €]ƒ ] J%#U(€]K(R€ƒ ] \*_:7"Bl ≤   <\$E.e&$[5$@iAj    yRD/S/(Y-D(^ €   =:>#?>?'>@A*%B%C?6D'( 6D € j= α €] ]= α €   j= α [...]... + c) + b2(c + a) + c2(a + b)=(a + b)(b + c)(c + a) 2abc = 82abc 6 Suy ra pcm ( ) 1.4 Bi tp : T 4.vip.pro.friendly lp 10 Toỏn 32 Truong THPT chuyờn Lờ Quý ụn Chuyờn Bt ng thc Bai 1: Cho a, b, c là các số dơng Chứng minh rằng: a3 b3 c3 a+b+c + 2 + 2 2 2 2 a + ab + b b + bc + c c + ca + a 3 Bai 2: Cho a,b,c 0 va a+b+c=3 Chng minh: a2 b2 c2 + + 1 a + 2b 2 b + 2c 2 c + 2a 2 Bai 3: Cho a,b,c 0 va . ++ ++++ ++ Cộng các BĐT này lại với nhau theo từng vế ta đợc BĐT cần chứng minh. 1.1.1.1. ab$)c"dEE <E"&$1$+>( Z` !a&$ 1$(>b$:/b !c!da/*$, '&$1$$9-I$F'(Z5&$1$9E+5 '(Z!/>b$eV%$0$E$a&$1$ =:>#?>?'>@A*%B%C?6D'( e  $EP!&$1$Cauchy