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培优点 5 隐零点
隐零点问题是指函数的零点存在但无法直接求解出来的问题，在函数、不等式与导数的
综合题目中常会遇到涉及隐零点的问题，处理隐零点问题的基本策略是判断单调性，合理取
点判断符号，再结合函数零点存在定理处理．
题型一 不含参函数的隐零点问题
例1 (2023·咸阳模拟)已知f(x)＝(x－1)2ex－x3＋ax(x>0)(a∈R)．
(1)讨论函数f(x)的单调性；
(2)当a＝0时，判定函数g(x)＝f(x)＋ln x－x2零点的个数，并说明理由．
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思维升华 已知不含参函数f(x)，导函数方程f′(x)＝0的根存在，却无法求出，利用函数零
点存在定理，判断零点存在，设方程f′(x)＝0的根为x ，则①有关系式f′(x)＝0成立，②
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注意确定x 的合适范围．
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跟踪训练1 (2023·天津模拟)已知函数f(x)＝ln x－ax＋1，g(x)＝x(ex－x)．
(1)若直线y＝2x与函数f(x)的图象相切，求实数a的值；
(2)当a＝－1时，求证：f(x)≤g(x)＋x2.
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题型二 含参函数的隐零点问题
例2 (2024·包头模拟)已知函数f(x)＝aex－ln(x＋1)－1.
(1)当a＝e时，求曲线y＝f(x)在点(0，f(0))处的切线与两坐标轴所围成的三角形的面积；
(2)证明：当a>1时，f(x)没有零点．
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跟踪训练2 已知函数f(x)＝ex－a－ln x＋x.(1)当a＝1时，求曲线f(x)在点(1，f(1))处的切线方程；
(2)当a≤0时，证明：f(x)>x＋2.
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1．(2023·荆门模拟)设函数f(x)＝ex＋bsin x，x∈(－π，＋∞)．若函数f(x)在(0，f(0))处的切
线的斜率为2.
(1)求实数b的值；
(2)求证：f(x)存在唯一的极小值点x，且f(x)>－1.
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2．(2023·绵阳模拟)已知函数f(x)＝ax－ln x，a∈R.
(1)若a＝，求函数f(x)的最小值及取得最小值时的x的值；
(2)若函数f(x)≤xex－(a＋1)ln x对x∈(0，＋∞)恒成立，求实数a的取值范围．
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