2024年数学高考大一轮复习第三章 §3.7　利用导数研究函数的零点

§3.7　利用导数研究函数的零点

考试要求　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 已知函数f(x)＝xsin x－1.

(1)讨论函数f(x)在区间上的单调性；

(2)证明：函数y＝f(x)在[0，π]上有两个零点．

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

思维升华 利用函数性质研究函数的零点，主要是根据函数单调性、奇偶性、最值或极值的符号确定函数零点的个数，此类问题在求解过程中可以通过数形结合的方法确定函数存在零点的条件．

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．

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

题型二　数形结合法研究函数的零点

例2 (2023·郑州质检)已知函数f(x)＝ex－ax＋2a，a∈R.

(1)讨论函数f(x)的单调性；

(2)求函数f(x)的零点个数．

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

思维升华 含参数的函数零点个数，可转化为方程解的个数，若能分离参数，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 (2023·长沙模拟)已知函数f(x)＝aln x－2.

(1)若a＝2，求曲线y＝f(x)在x＝1处的切线方程；

(2)若函数f(x)在(0,16]上有两个零点，求a的取值范围．

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

题型三　构造函数法研究函数的零点

例3 (12分)(2022·全国乙卷)已知函数f(x)＝ln(1＋x)＋axe－x.

(1)当a＝1时，求曲线y＝f(x)在点(0，f(0))处的切线方程；[切入点：求f′(x)，f′(0)]

(2)若f(x)在区间(－1,0)，(0，＋∞)各恰有一个零点，求a的取值范围．

[关键点：根据导数f′x对a分类讨论，对x分－1，0与0，＋∞,两部分]

思维升华 涉及函数的零点(方程的根)问题，主要利用导数确定函数的单调区间和极值点，根据函数零点的个数寻找函数在给定区间内的极值以及区间端点的函数值与0的关系，从而求得参数的取值范围．

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．

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________