竞赛专题--凸函数和琴生不等式

凸函数和琴生不等式

1..(02成都模拟试题)若函数y?sinx在区间(0,?)上是凸函数，那么在?ABC中，sinA?sinB?sinC的最大值为A

32313 B C D

2222分析：

?y?sinx在(0,?)上是凹函数，则：1A?B?C3(sinA?sinB?sinC)?sin()?sin60??33233sinA?sinB?sinC?2当且仅当sinA?sinB?sinC时，即A?B?C??3时，取等号；2.若a1,a2,?an是一组实数，且a1?a2???an?k(k为定值)，试求：a1?a2???an的最小值分析：?f(x)?x2在(??,??)上是凸函数a1?a2???an2k21222?(a1?a2???an)?()?2nnnk2222?a1?a2???an?n当且仅当a1?a2???an时，取等号2223.已知xi?0,(i?1,2,?,n)，n?2,x1?x2???xn?1，求证：(1?1n11)?(1?)n???(1?)n?n(n?1)nx1x2xn1111111证：?[(1?)n?(1?)n???(1?)n]?n(1?)n(1?)n?(1?)nnx1x2xnx1x2xn

?(1?1n111)(1?)?(1?)x1x2xn1n

bbb1bbb)(1?2)?(1?n)]?1?(12?n));a1a2ana1a2an1111111?[(1?)(1?)?(1?)]n?1?()n?1?nxx?xx1x2xnx1x2?xn12n(利用结论：[(1?又?nx1x2?xn?x1?x2???xn1?nn

111?[(1?)(1?)?(1?)]n?1?nx1x2xn?(1?(1?

1111)(1?)?(1?)?(n?1)nx1x2xn1n11)?(1?)n???(1?)n?n(n?1)nx1x2xn

4.若P为?ABC内任一点，求证?PAB、?PBC、?PCA中至少有一个小于或等于30?；证：设?PAB??、?PBC??、?PCA??，且?PAC??'、?PBA??'、?PCB??';PAsin??PBsin?'??依正弦定理有：PBsin??PCsin?'??sin?sin?sin??sin?'sin?'sin?'PCsin??PAsin?'???(sin?sin?sin?)2?sin?sin?sin?sin?'sin?'sin?'

sin??sin??sin??sin?'?sin?'?sin?'6)6???????'??'??'1?sin6()?()662?(1?sin?sin?sin??()321

2???30?,否则??150?时，?、?中必有一个满足??30??在?、?、?,中必有一个角满足sin??

补充练习：

1.若xi?R(1?i?n),?xi?1，求证：(x1??i?1n1111)(x2?)?(xn?)?(n?)n; x1x2xnn2xy;

2.已知x?0,y?0,x2?y2?1，求证:x3?y3?33.A、B、C为?ABC的三个内角，求证：cosA?cosB?cosC?;

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