2023版考前三个月冲刺专题练　第10练　零点问题【无答案版】

这是一份2023版考前三个月冲刺专题练　第10练　零点问题【无答案版】，共3页。
第10练　零点问题[考情分析]　在近几年的高考中，函数与方程、不等式的交汇是考查的热点，常以指数函数、对数函数以及三角函数为载体考查函数的零点(方程的根)问题，难度较大，多以压轴题出现． 一、 判断零点个数问题例1　(2019·全国Ⅰ)已知函数f(x)＝sin x－ln(1＋x)，f′(x)为f(x)的导数．证明：(1)f′(x)在区间上存在唯一极大值点；(2)f(x)有且仅有2个零点．______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________规律方法　利用导数研究函数的零点(1)如果函数中没有参数，一阶导数求出函数的极值点，判断极值点大于0、小于0的情况，进而判断函数零点个数．(2)如果函数中含有参数，往往一阶导数的正负不好判断，先对参数进行分类，再判断导数的符号，如果分类也不好判断，那么需要二次求导，判断二阶导数的正负时，也可能需要分类．跟踪训练1　(2022·浙江精诚联盟联考)已知函数f(x)＝ex－asin x，g(x)＝ln(x＋1)－asin x.(1)若y＝f(x)在上单调递增，求实数a的取值范围；(2)若不等式f(x)≥cos x在(－1，＋∞)上恒成立，判断函数g(x)在(－1,1)上的零点个数，并说明理由．______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________二、由零点个数求参数范围例2　(2022·全国乙卷)已知函数f(x)＝ax－－(a＋1)ln x.(1)当a＝0时，求f(x)的最大值；(2)若f(x)恰有一个零点，求a的取值范围．______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________规律方法　已知零点个数求参数范围时(1)根据区间上零点的个数估计函数图象的大致形状，从而推导出导数需要满足的条件，进而求出参数满足的条件．(2)也可以先求导，通过求导分析函数的单调性，再依据函数在区间内的零点情况，推导出函数本身需要满足的条件，此时，由于函数比较复杂，常常需要构造新函数，通过多次求导，层层推理得解．跟踪训练2　(2022·烟台模拟)已知函数f(x)＝ln x－ax＋a(a∈R)．(1)讨论f(x)的单调性；(2)若f(x)在(1，＋∞)上有零点x0，①求a的取值范围；②求证：<x0<.___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________