1、22222,()()()(1a b c dRabcdacbdadbc 定定理理（二二维维形形式式的的柯柯西西不不等等若若，则则当当且且仅仅当当时时，式式）等等号号成成立立）：平平方方和和的的积积简简记记积积的的和和的的平平方方1231231232222223121212312311223323,0()()()aaaR bbbRb b baaabbba ba baaabba bb 当当若若且且，三三维维柯柯西西不不等等且且仅仅当当时时，等等则则（式式：号号成成立立）123123(,),(,),|aa aabb b ba babrrrrr rrrr rrr何何意意义义则则几几：若若:用用柯柯证证法
2、法一一西西不不等等式式149,6,1,36x y zRxyzxyz 例例 已已知知且且求求证证22211111,.49632xyzxyz当当且且仅仅当当即即时时取取等等号号14936xyz1,xyz又又2149123()()()xyzxyzxyzxyz 1112,3,.632yx zxxyz当当且且仅仅当当即即时时取取等等号号14461236494914()()()yxzxzyxyxzyz149149()()xyzxyzxyz :证证法法二二 逆逆代代法法例例1.222231xyzxyz例例已已知知，求求的的最最小小值值；222,32621x y zxyzpxyz 已已知知满满足足，求求变变式
3、式：的的最最大大值值；21(,)3224243.22xxxx 已已 知知，求求 变变：的的 最最 大大 值值式式222,(24)(1)(44).x y zRxyxxy变变式式3 3已已知知，求求的的最最小小值值：例例2.2222,1.1(1);2224(093(2)444xyzx y zxyzxyzyzzxxy已已知知正正数数满满例例足足浙浙江江自自选选模模块块)求求证证:求求的的最最小小值值.猜想柯西不等式的一般形式猜想柯西不等式的一般形式，aaaAn22221 设设，bbbCn22221 nnbababaB 22112ACB不不等等式式就就是是分析：分析：)()(2)()(22221221
4、1222221nnnnbbbxbababaxaaaxf 构造二次函数构造二次函数0)()()()(2222211 nnbxabxabxaxf又又。等等号号成成立立时时使使得得或或存存在在一一个个数数当当且且仅仅当当则则是是实实数数设设一一般般形形式式的的柯柯西西不不等等式式定定理理,),2,1(,),2,1(0,)(321321nikbaknibbbbbaaaaiiinn 同样这个不等式也有着向量（同样这个不等式也有着向量（n维向量）及几何背景，维向量）及几何背景，其应用广泛。其应用广泛。22212122212121221212122212(1)()111 (111)(11 )(11111 1
5、)()11nnnnnnnnnxxxnxxxxxxxxxxxxxxxxxxxxxxxx 证明：证明：122221212122.,.111nnnna aaRaaaaaannnaaa 例例 已已知知，证证明明：222222222111,01111127.1932abcabba b cabcbcaabcaccabc已已知知，；且且，则则；例例；：练习练习:分式型：分母和非常数，分式型：分母和非常数，但具有轮换特征但具有轮换特征分式型：分母和非常数，分式型：分母和非常数，但具有轮换特征但具有轮换特征其其 他他 根据上面结果，你能猜想出一般形式的柯西不根据上面结果，你能猜想出一般形式的柯西不等式吗？等式吗
6、？1231231232222223121212312311223323,0()()()a aaR b b bRb b baaabbba ba baaabba bb 当当若若且且，三三维维柯柯西西不不等等且且仅仅当当时时，等等则则（式式：号号成成立立）猜想柯西不等式的一般形式猜想柯西不等式的一般形式，aaaAn22221 设设，bbbCn22221 nnbababaB 22112ACB不不等等式式就就是是分析：分析：)()(2)()(222212211222221nnnnbbbxbababaxaaaxf 构造二次函数构造二次函数0)()()()(2222211 nnbxabxabxaxf又又。等
7、等号号成成立立时时使使得得或或存存在在一一个个数数当当且且仅仅当当则则是是实实数数设设一一般般形形式式的的柯柯西西不不等等式式定定理理,),2,1(,),2,1(0,)(321321nikbaknibbbbbaaaaiiinn 同样这个不等式也有着向量（同样这个不等式也有着向量（n维向量）及几何背景，维向量）及几何背景，其应用广泛。其应用广泛。继续继续2答案答案22221212()()nnn aaaaaa222212121()nnaaaaaan补充练习补充练习22212122212121221212122212(1)()111 (111)(11 )(11111 1)()11nnnnnnnnnx
8、xxnxxxxxxxxxxxxxxxxxxxxxxxx 222121211111nnxxxxxxn证明：证明：22222222222222:4()(1111)()()4(16)(8),6446416165160,05abcdabcdabcdeeeeeeee 解解即即即即故故 2答案答案团Tiffany，a 16yearold girl，was very shy.Last September，her best frien“I was really sad the moment I heard the bad news and I didnt know what to do，”Tiffany re
9、called.“I shut myself in my room for a whole week.It was then that my aunt took me to a sports club one Saturday and I saw so many young people playing different kinds of sports there.I signed up for a beginners course in volleyball and since then I have been playing this sport.Now I practice twice
10、a week there.It is wonderful playing sports in this club and I have made lots of friends as well.2 ”The most basic aim of playing sports is that you can improve your health even if you are not very good at sports.Besides，you can get to know a circle of people at your age while playing sports.3 Since
11、 she joined the sports club，s I got used to the life here.And now I know lots of(5)_ here.For example,when I meet my friend on the street,I usually(6)_ him like this,“Hey,where are you going?”In our country if someone asks this,people may get(7)_ but in this country people wont.Of course,there are s
12、ome other interesting things here.Ill tell you about them next time.he has opened up herself and now she has become very active and enjoys meeting and talking with others.1Its polite for girls to kiss each other on the side of the face.s also become more confident.团圆圆一家在台湾可受欢迎了。每天，小朋友们排着长队，等着跟它们合影留念。从“排着长队”体现出每天喜欢它们的人不计其数，特别受选D.A.根据同类项合并法则,与不是同类项,不能合并,故本选项错误;B.根据算术平方根的定义,=3,故本选项错误;C.根据同底数幂的乘法aa2=a3,故本选项错误;D.根据积的乘方,(2a3)2=4a6,故本选项正确.欢迎。从“合影留念”体现出大家都想和大熊猫留住最美丽的瞬间以作纪念。Nothing can be accomplished without norms or standards.精品资料！感谢阅读下载！
