专题1　培优点4　极值点偏移问题_2.2025数学总复习_2023年新高考资料_二轮复习_2023年高考数学二轮复习讲义+课件（新高考版）_2023年高考数学二轮复习讲义（新高考版）_学生版_730

培优点 4 极值点偏移问题 极值点偏移是指函数在极值点左右的增减速度不一样，导致函数图象不具有对称性，极 值点偏移问题常常出现在高考数学的压轴题中，这类题往往对思维要求较高，过程较为烦琐， 计算量较大，解决极值点偏移问题，有对称化构造函数法和比值代换法，二者各有千秋，独 具特色． 考点一 对称化构造函数 例1 (2022·全国甲卷)已知函数f(x)＝－ln x＋x－a. (1)若f(x)≥0，求a的取值范围； ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ (2)证明：若f(x)有两个零点x，x，则xx<1. 1 2 1 2 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 规律方法 对称化构造法构造辅助函数：对结论 x ＋x>2x 型，构造函数F(x)＝f(x)－f(2x － 1 2 0 0 x)；对结论xx>x型，方法一是构造函数F(x)＝f(x)－f ，通过研究F(x)的单调性获得不等式； 1 2 方法二是两边取对数，转化成ln x＋ln x>2ln x，再把ln x，ln x 看成两变量即可． 1 2 0 1 2 跟踪演练1 已知函数f(x)＝＋ln x. (1)求f(x)的极值和单调区间； (2)若函数g(x)＝f(x)－a(a>2)的两个零点为x，x，证明：x＋x>4. 1 2 1 2 ________________________________________________________________________________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 考点二 比值代换 例2 (2022·六安模拟)已知函数f(x)＝xln x－ax2＋x(a∈R)．若f(x)有两个零点x ，x ，且 1 2 x>2x，证明：xx>. 2 1 1 2 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 规律方法 比值代换法是指通过代数变形将所证的双变量不等式通过代换 t＝化为单变量的 函数不等式，利用函数单调性证明． 跟踪演练2 (2022·湖北圆创联考)已知f(x)＝x2－2aln x，a∈R.若y＝f(x)有两个零点x ， 1 x(x4x. 0 1 2 0 ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________