(满分13分)已知函数
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(1)求
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的单调区间;
(2)记
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在区间
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上的最小值为
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令
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;
①如果对一切n,不等式
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恒成立,求实数c的取值范围;
②求证:
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。
解:(I)因为
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,所以函数定义域为
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,且
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。
由
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得
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,
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的单调递增区间为
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;
由
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<0得
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,
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的单调递增区间为(0,+
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).
(II) 因为
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在
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上是减函数,所以
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则
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.
①:
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>
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又lim
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,
因此
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,即实数
c的取值范围是
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.
② : 由① 知
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③
因为[
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]
2
所以
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<
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(n
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N
*),
则
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<
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
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
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
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
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
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
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
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
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
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
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中,公差
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,
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