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在我们平时教学中,学生做错练习题是常见的,但主动寻找错误原因的同学还很不多。在解题过程中,对错误解法进行分析,找出病因,对巩固基础知识,提高解题能力是非常必要的。下面仅就一道习题几种常见错误解法进行剖析,并提出正确的解法,供参考。题目设x、y为正变数,a、b为正常数,且a/x+b/y=1,求x+y的最小值。错解一∵a、b、x,y为正数,∴a/x及b/y均为正数,∴a/x+b/y≥2((ab/xy)~(1/2)),而a/x+b/y=1.∴(ab/xy)~(1/2)≤1/2.∴(xy/ab)~(1/2)≥2∴xy~(1/2)≥2((ab)~(1/2)),又∵x+y≥2((xy)~(1/2))∴x+y≥4((ab)~(1/2)),∴x+y的最小值为4(ab)~(1/2)
In our usual teaching, it is common for students to do wrong exercises, but there are not many students who actively look for the reasons for the errors. In the process of solving the problem, it is necessary to analyze the wrong solution method to find out the cause of the disease, and to consolidate the basic knowledge and improve the problem-solving ability. In the following, we will analyze only a few common mistakes in one problem and propose correct solutions for reference. The title sets x and y as positive variables, a and b are normal numbers, and a/x+b/y=1. Find the minimum value of x+y. A, b, x, y are positive numbers, ∴a/x and b/y are positive numbers, and ∴a/x+b/y≥2((ab/xy)~(1/2) ), and a/x+b/y=1.∴(ab/xy)~(1/2)≤1/2.∴(xy/ab)~(1/2)≥2∴xy~(1/ 2) ≥2((ab)~(1/2)) and ∵x+y≥2((xy)~(1/2))∴x+y≥4((ab)~(1/2) ), The minimum value of ∴x+y is 4(ab)~(1/2)