专题1第6讲母题突破1　导数与不等式的证明_2.2025数学总复习_赠品通用版（老高考）复习资料_二轮复习_2023年高考数学二轮复习讲义+课件（全国版理科）_学生版_学生用书Word版文档_857

第 6 讲 导数的综合应用 [考情分析] 1.利用导数研究函数的单调性与极值(最值)是高考的常见题型，而导数与函数、 不等式、方程、数列等的交汇命题是高考的热点和难点.2.多以解答题的形式压轴出现，难度 较大． 母题突破 1 导数与不等式的证明 母题 已知函数f(x)＝ex－x2. (1)求曲线f(x)在x＝1处的切线方程； (2)求证：当x>0时，≥ln x＋1. 思路分析 ❶求切线方程 ↓ ❷fx≥e－2x＋1 ↓ ❸ex－x2－e－2x－1≥0 ↓ ❹ex＋2－ex－1≥x2 ↓ ❺≥x≥ln x＋1 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ [子题1] 已知函数f(x)＝ex－ax－a，当a＝1时，令g(x)＝.求证：当x>0时，g(x)<1. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ [子题2] (2022·德州联考改编)已知函数f(x)＝ln .若x∈(0,1)，求证：f(x)g(x)(或 f(x)0(或 f(x)－ g(x)<0)，进而构造辅助函数h(x)＝f(x)－g(x)． (2)适当放缩构造法：一是根据已知条件适当放缩；二是利用常见放缩结论． (3)构造“形似”函数，稍作变形再构造，对原不等式同结构变形，根据相似结构构造辅助函 数． 1．(2021·全国乙卷改编)设函数f(x)＝ln(1－x)，函数g(x)＝.求证：g(x)<1. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 2．已知函数f(x)＝ex－a－ln(x＋a)．当a≤1时，证明：f(x)>0. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________