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题目:(《数学通报》2012年第9期)已知正数a、b、c满足a+2b+3c≤abc,求5a+22b+c的最小值。其解答是通过对条件和目标式子变形并经过巧妙的拆分,再利用三元均值不等式求得最小值。但该方法变形、拆分的技巧性较强且不易被学生掌握。本文通过换元、待定系数等手段,结合均值不等式得出一个更利于学生掌握的一般解法,而且利用本方法也可轻松解决2008年全国高中数学联赛吉林省预赛第17题。
Title: The positive numbers a, b, c satisfy a + 2b + 3c≤abc and find the minimum value of 5a + 22b + c. The answer is to find the minimum value by using the ternary mean inequality by deforming the condition and the target formula and cleverly splitting them. However, the method is distorted and the technique of splitting is strong and not easily mastered by students. In this paper, we get a general solution that is more conducive to students' mastery by means of changing yuan, undetermined coefficients and mean inequality, and also can easily solve the 17th Jilin Province Preliminary Round of the 2008 National Senior High School Maths Match by using this method.