已知函數(shù)
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和點

,過點

作曲線

的兩條切線
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、
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,切點分別為
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、

.
(1)求證:

為關(guān)于

的方程

的兩根;
(2)設(shè)

,求函數(shù)

的表達式;
(3)在(2)的條件下,若在區(qū)間
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內(nèi)總存在

個實數(shù)
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(可以相同),使得不等,則m的最大值,
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為正整數(shù)
解:(1)由題意可知:
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∵
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,
∴切線

的方程為:

,
又
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切線

過點
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,
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有
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,
即

, ①
同理,由切線

也過點

,得

.②
由①、②,可得

是方程

( * )的兩根
(2)由( * )知.




,
∴

.
(3)易知

在區(qū)間

上為增函數(shù),



,
則

.
即

,即

,
所以

,由于

為正整數(shù),所以

.
又當

時,存在

,

滿足條件,
所以

的最大值為

.
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
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
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