本试题主要考查了导数的概念和导数的运算,以及导数的几何意义的运用,并利用导数研究函数的单调性和函数的零点问题的综合运用试题。
(1)先求解导数,利用点斜式写出切线方程。
(2)原方程等价于
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,令
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则函数
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与
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在
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轴右侧有唯一交点。转化为图像与图像的交点来处理。
(3)分别分析函数的单调区间,然后结合结论,判定都是单调增函数时的参数的取值范围
解:(1)
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; ……………3分
(2)原方程等价于
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,令
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则函数
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与
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在
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轴右侧有唯一交点。
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当
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或
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时
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,当
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时
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在
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上单调递减,在
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上单调递增。
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时有极小值
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,
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时有极大值
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当
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有唯一解时
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或
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……………8分
(3)
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,
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当
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时
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,当
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时
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在
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上单调递减,在
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上单调递增。
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在
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上单调递减,在
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上单调递增。
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与
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在
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上单调递增, 使得
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与
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在
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上均为增函数则满足
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,不等式组无解,故不存在实数