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利用不等式求极值,是解决极值问题的一个重要的方法。其根据就是:若干个非负实数的算术平均值不小于其几何平均值,仅在这些非负实数都相等时,算术平均值才等于几何平均值。即:若x_1,x_2,x_3,…,x_n非负,n>1,则(仅在x_1=x_2=x_3…=x_n时,等号成立。) 因此,在这若干个非负实数相等时,如果这若干个非负实数之积一定,则和最小;和一定,则积最大。现试举二例以说明此结论之应用。
Using inequality to find the extreme value is an important way to solve the extreme value problem. The rationale is that the arithmetic mean of a number of nonnegative real numbers is not less than its geometric mean, and the arithmetic mean is equal to the geometric mean only if these nonnegative real numbers are equal. That is, if x_1, x_2, x_3, ..., x_n are nonnegative and n> 1, (Equations are valid only when x_1 = x_2 = x_3 ... = x_n.) Therefore, when several nonnegative real numbers are equal, If the product of some non-negative real number of a certain, then and minimum; and certainly, the product of the largest. Now give two examples to illustrate the application of this conclusion.