真题
1 . 已知函数f(x)=lnx﹣x+1,x∈(0,+∞),求函数f(x)的最大值;
(2)设a1,b1(k=1,2…,n)均为正数,证明:
①若a1b1+a2b2+…anbn≤b1+b2+…bn,则![](https://img.xkw.com/dksih/QBM/2014/5/30/1571743289548800/1571743295283200/STEM/49f34411c0ef4df98faed98205fe32a7.png)
…
≤1;
②若b1+b2+…bn=1,则
≤![](https://img.xkw.com/dksih/QBM/2014/5/30/1571743289548800/1571743295283200/STEM/fe1977caa7284a9d8ca9d900f60ed1d2.png)
…
≤b12+b22+…+bn2.
(2)设a1,b1(k=1,2…,n)均为正数,证明:
①若a1b1+a2b2+…anbn≤b1+b2+…bn,则
![](https://img.xkw.com/dksih/QBM/2014/5/30/1571743289548800/1571743295283200/STEM/49f34411c0ef4df98faed98205fe32a7.png)
![](https://img.xkw.com/dksih/QBM/2014/5/30/1571743289548800/1571743295283200/STEM/d0477e4c484d45b79c42365f68b099e6.png)
![](https://img.xkw.com/dksih/QBM/2014/5/30/1571743289548800/1571743295283200/STEM/db122dcbf85441ab8caaaaf39e892c83.png)
②若b1+b2+…bn=1,则
![](https://img.xkw.com/dksih/QBM/2014/5/30/1571743289548800/1571743295283200/STEM/4da1c0450d4b42f1a905e79b2f1dca33.png)
![](https://img.xkw.com/dksih/QBM/2014/5/30/1571743289548800/1571743295283200/STEM/fe1977caa7284a9d8ca9d900f60ed1d2.png)
![](https://img.xkw.com/dksih/QBM/2014/5/30/1571743289548800/1571743295283200/STEM/e8315771c82b425a883f31b0f2215985.png)
![](https://img.xkw.com/dksih/QBM/2014/5/30/1571743289548800/1571743295283200/STEM/5b97963255eb403d90413d0998b6a3d5.png)
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真题
2 . 设a>0,b>0,已知函数f(x)=
.
(1)当a≠b时,讨论函数f(x)的单调性;
(2)当x>0时,称f(x)为a、b关于x的加权平均数.
(1)判断f(1),f(
),f(
)是否成等比数列,并证明f(
)≤f(
);
(2)a、b的几何平均数记为G.称
为a、b的调和平均数,记为H.若H≤f(x)≤G,求x的取值范围.
![](https://img.xkw.com/dksih/QBM/2014/5/22/1571735315456000/1571735321034752/STEM/a887b364c9ae430085b5453fd761f245.png)
(1)当a≠b时,讨论函数f(x)的单调性;
(2)当x>0时,称f(x)为a、b关于x的加权平均数.
(1)判断f(1),f(
![](https://img.xkw.com/dksih/QBM/2014/5/22/1571735315456000/1571735321034752/STEM/f5ccaae06a0846a2b94036f8707b09a6.png)
![](https://img.xkw.com/dksih/QBM/2014/5/22/1571735315456000/1571735321034752/STEM/e8c6fe1321e7462c84f7d149db7859f0.png)
![](https://img.xkw.com/dksih/QBM/2014/5/22/1571735315456000/1571735321034752/STEM/e8c6fe1321e7462c84f7d149db7859f0.png)
![](https://img.xkw.com/dksih/QBM/2014/5/22/1571735315456000/1571735321034752/STEM/f5ccaae06a0846a2b94036f8707b09a6.png)
(2)a、b的几何平均数记为G.称
![](https://img.xkw.com/dksih/QBM/2014/5/22/1571735315456000/1571735321034752/STEM/7a8b7cdc6da74f19aeacc9bd46f7d1b0.png)
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