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§3.7 利用导数研究函数的零点
课标要求 函数零点问题在高考中占有很重要的地位，主要涉及判断函数零点的个数或范
围．高考常考查三次函数与复合函数的零点问题，以及函数零点与其他知识的交汇问题，一
般作为解答题的压轴题出现．
题型一 利用函数性质研究函数的零点
例1 (2023·辽宁实验中学模拟)已知函数f(x)＝excos x.
(1)求f(x)在区间内的极大值；
(2)令函数h(x)＝－1，当a>时，证明：h(x)在区间内有且仅有两个零点．
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跟踪训练1 (2023·芜湖模拟)已知函数f(x)＝ax＋(a－1)ln x＋－2，a∈R.
(1)讨论f(x)的单调性；
(2)若f(x)只有一个零点，求a的取值范围．
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题型二 数形结合法研究函数的零点
例2 (2023·安庆模拟)已知函数f(x)＝aln x＋bx2e1－x，a，b∈R.e＝2.718 28….
(1)若曲线y＝f(x)在点(2，f(2))处的切线方程是y＝x＋ln 2，求a和b的值；
(2)若a＝e，讨论导函数f′(x)的零点个数．
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思维升华 含参数的函数的零点个数，可转化为方程解的个数，若能分离参数，则可将参数
分离出来后，用x表示参数的函数，作出该函数的图象，根据图象特征求参数的范围或判断
零点个数．跟踪训练2 (2024·厦门模拟)设函数f(x)＝ln x－ax2－bx(a，b∈R)．
(1)当a＝2，b＝1时，求函数f(x)的单调区间；
(2)当a＝0，b＝－1时，方程f(x)＝mx在区间[1，e2]上有唯一实数解，求实数m的取值范围．
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题型三 构造函数法研究函数的零点
例3 已知函数f(x)＝ex＋x＋4ln(2－x)．
(1)求函数f(x)的图象在点(0，f(0))处的切线方程；
(2)判断函数f(x)的零点个数，并说明理由．
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跟踪训练3 (2021·全国甲卷)已知a>0且a≠1，函数f(x)＝(x>0)．
(1)当a＝2时，求f(x)的单调区间；
(2)若曲线y＝f(x)与直线y＝1有且仅有两个交点，求a的取值范围．
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