高中数学-公式-柯西不等式

1、精品文档 ac d( )( )( )ac a2b2c2d22 d( )( )( ) .a2b2c2d22( )( )( ) =.=( ) 0a2b2c2d2ac bd2ad 2( )2c ba2b2c2d2a2c2a2d2b22d2(acbd) (ad bc) (acbd) . 222rr ( , )， ( , )|m a b|n c d .，ma b n c d2222 rm n r r r r r g | |ncosm,n | g | | |. m n m nm n m ( )( ) ) f xa2b2x2 x c2d2f(x)(axc)bxd)0.22 2( ) 4( )( )ac bd

2、2a2b2c2d2a2 b2 g c2 d2 | ,a2 b2 c2 d2 g |或 g.a2 b2 c2 d2 ac bd或 | | |. ） ,(ac) bd) . d da2b2c2222 , , , x y x y Rx21y21x22y2(x x ) (y y ) .22112 221212 , , , , , x y x y x y R112233 ：(a b )(c d )(acbd)x y x y (x x ) (y y )；2222221212222221212 x1 2 ?yx| a b g c d .2222 3 1 102 yxx 3 1 102 ,( , , , ,

3、, )y a bx c d e fx a b c d e f R yxx 3x2y1x y2.2111 ( )(3 2 ) (3 2 ) .x2y2x2y222xy2131313 1 1 , ， 2.2x y Rx yx y 1 1 11 1x y111 (x y ) x) ( y) ) () 2222x y 22xy.精品文档b R a b1 1 、 ( )( )4.aa b x y a b R x ya b , , , 1 .x ya bx y x y ( ).x y若 , , 1x y z . 2x y z R x y z22 , , (a b )(c d )(acbd) (a b c

4、)(d e f )( cf)1x y z .x y z R x y z；222222222222 | | |nna a a b b b R , ,L , , , ,L , 12n12n(a a L a b b L b )(ab a b a b )2122212222n2a an1 122n nab L 0）b1b b2ni12n B ab a bab ， A aL，Cb b L b 2 0B AC 21a22a2212221 122nnnnn f aa a)x2(ab a b a b )x bb b )22222221221n1 122nnnf(x)(a xb )(a xb )+a xb )0

5、.2221122nna a 0又a22212n 2(ab a b a b ) 2 4(a a L a )gb b L b )02222221 12 2n n12n12n. ）1aL ( ) . 21a22an2a aa2n12n 3x2yz1x y z.222 1 1 1y z , , 1 .x y z Rx2 3x y z114 a b c.ab bc ac1111( )(a ca b b c)( )( )()(11) 42ab bcab bc: ; . , , 是 , , a a a b b b c c c b bbn12n12n12n12ab a b +)a b1 12 2nna cac a c +1 12 2nna bab a b +1 n2n1n 1.精品文档 = 或 = a a a b bb12n12n , , 是na aa12n1 11aaann21 a .223322 3n21 设 , , 是 , , 2, .b bba a1ab bbb1b2b n12n2n12nn111又1 22 2n2aaabbba b 22332nn223nn22232112. a,b,c2(a b c