已知函數(shù)
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(1)若函數(shù)
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的圖象切x軸于點(diǎn)(2,0),求a、b的值;
(2)設(shè)函數(shù)

的圖象上任意一點(diǎn)的切線斜率為k,試求
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的充要條件;
(3)若函數(shù)

的圖象上任意不同的兩點(diǎn)的連線的斜率小于l,求證
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.
試題分析:(1)由函數(shù)
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的圖象切x軸于點(diǎn)(2,0),得
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且
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,解方程組可得
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的值.
(2)由于
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,根據(jù)導(dǎo)數(shù)的幾何意義,任意不同的兩點(diǎn)的連線的斜率小于l,
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對(duì)任意的
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恒成立,利用分離變量法,轉(zhuǎn)化為
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對(duì)任意的
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恒成立,進(jìn)一步轉(zhuǎn)化為函數(shù)的最值問題;
(3)設(shè)
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,則
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對(duì)
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恒成立
將上不等式看成是關(guān)于
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的一元二次不等式即可.
試題解析:解:(1)
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由
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,得
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,
又
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,得
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(2)
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對(duì)任意的
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,即
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對(duì)任意的
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恒成立
等價(jià)于
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對(duì)任意的
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恒成立
令
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則
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,當(dāng)且僅當(dāng)
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時(shí)“=”成立,
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
在
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上為增函數(shù),
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(3)設(shè)
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,則
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即
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,對(duì)
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恒成立
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,對(duì)
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恒成立
即
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,對(duì)
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恒成立
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解得
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