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在学习函数、方程、不等式过程中,常见到“恒成立”问题.一般来说,“恒成立”问题多数涉及两个变量,其中一个变量恒满足某一条件,对另一个变量进行数学设问.而这两个变量间的关系常以函数、方程、不等式等形式给出.本文重点从函数角度介绍一下“恒成立”问题的解题策略. 一、不等式“恒成立”问题 例1 已知x2+(4a-3)x+3a>0, (1)若不等式对任意实数x∈[-1,3]恒成立,求实数a的取值范围. (2)若不等式对任意实数a∈[-1,3]恒成立,求实数x的取值范围.
In the process of learning functions, equations, and inequalities, the problem of “constant establishment” is commonly observed. In general, most of the “constant establishment” problems involve two variables. One of the variables always satisfies a certain condition, and the other variable is mathematically asked. The relationship between these two variables is often given in the form of functions, equations, inequalities, etc. This article focuses on the problem-solving strategies of the “constant establishment” problem from the perspective of functions. I. Inequality “constant establishment” problem Example 1 Known X2+(4a-3)x+3a>0, (1) If the inequality holds for any real number x∈[-1,3], find the range of real numbers a. (2) If the inequality holds for any real number a∈[ -1,3] is established and the range of values of the real number x is obtained.