試題分析:本題主要考查導(dǎo)數(shù)的運(yùn)算、利用導(dǎo)數(shù)求曲線的切線、利用導(dǎo)數(shù)判斷函數(shù)的單調(diào)性、利用導(dǎo)數(shù)求函數(shù)的最值、恒成立問題等基礎(chǔ)知識,考查學(xué)生的分析問題解決問題的能力、轉(zhuǎn)化能力、計(jì)算能力,考查學(xué)生的分類討論思想、函數(shù)思想.第一問,對
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求導(dǎo),將切點(diǎn)的橫坐標(biāo)代入得到切線的斜率,再將切點(diǎn)的橫坐標(biāo)代入到
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中,得到切點(diǎn)的縱坐標(biāo),利用點(diǎn)斜式得到切線的方程;第二問,
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在定義域
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內(nèi)是增函數(shù),只需
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在
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恒成立,對
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求導(dǎo),由于分母恒正,只需分子
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在
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恒成立,設(shè)函數(shù)
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,利用拋物線的性質(zhì)求出
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,令
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即可,解出P的值;第三問,先通過函數(shù)
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的單調(diào)性求出
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的值域,通過對P的討論研究
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的單調(diào)性,求出
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的值域,看是否有值大于
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的最小值為2.
(1)當(dāng)
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時,函數(shù)
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,
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.
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,曲線
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在點(diǎn)
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處的切線的斜率為
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.
從而曲線
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在點(diǎn)
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處的切線方程為
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,即
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.…4分
(2)
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.
令
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,要使
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在定義域
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內(nèi)是增函數(shù),只需
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在
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內(nèi)恒成立.
由題意
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,
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的圖象為開口向上的拋物線,對稱軸方程為
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,∴
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, 只需
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,即
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時,
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∴
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在
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內(nèi)為增函數(shù),正實(shí)數(shù)
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的取值范圍是
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.……9分
(3)∵
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在
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上是減函數(shù),
∴
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時,
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;
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時,
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,即
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,
①當(dāng)
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時,
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,其圖象為開口向下的拋物線,對稱軸
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在
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軸的左側(cè),且
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,所以
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在
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內(nèi)是減函數(shù).
當(dāng)
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時,
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,因?yàn)?img src="http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824052718947306.png" style="vertical-align:middle;" />
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,所以
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,
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,
此時,
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在
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內(nèi)是減函數(shù).
故當(dāng)
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時,
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在
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上單調(diào)遞減
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,不合題意;
②當(dāng)
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時,由
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,所以
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.
又由(2)知當(dāng)
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時,
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在
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上是增函數(shù),
∴
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,不合題意;
③當(dāng)
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時,由(2)知
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在
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上是增函數(shù),
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,
又
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在
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上是減函數(shù),故只需
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,
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,
而
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,
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,
即
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,解得
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,
所以實(shí)數(shù)
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的取值范圍是
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. 14分