一个二元问题的探究

• 作 者: 张寅历 查正开
• 机 构: 江苏省常熟中学高三(14)班,
• 出 处: 数学通讯
• 发表日期: 2017年07期
• 关键词: 柯西不等式 换元 问题结构 当且仅当 幂平均不等式 竞赛试题 可证

问题已知x,y∈R~+满足x/1+y+y/1+x,求x+y=2的最小值.这个问题结构简洁,形式优美,解法多样,值得加以研究.此题采用直接通分或利用均值换元都不难给予解决,但若要用柯西不等式处理则需一定的技巧,解答如下.因为x,y∈R~+,x+y=2,所以,由柯西不等式得[y(1+x)+x(1+y)]·(x/1+y+y/1+x)≥(x+y)~2.

Problem Known x, y ∈ R ~ + satisfies the minimum of x + 1 + y + y / 1 + x, and satisfies x + y = 2. The problem is simple in structure and graceful in form and solution. It is worth to be studied. It is not difficult to solve the problem by using the direct pass or the mean exchange. However, if we need to deal with the Cauchy inequality, we need certain skills. The solution is as follows: Because x, y∈R ~ +, x + y = 2, Cauchy inequality has [y (1 + x) + x (1 + y)] · (x / 1 + y + y / 1 + x) ≥ (x + y)