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两个正数的均值定理是高中数学的必修内容,在不等式证明和代数式求最值中经常用到,因此要求同学们熟练掌握.首先,两个正数的均值定理是指:如果a、b∈(0,+∞),那么a+b/2≥ab~(1/2),当且仅当a=b时等号成立.其内容通常可概括为:两个正实数的算术平均值((a+b)/2)不小于它们的几何平均值(ab~(1/2)),其次,由均值定理可得:两个正数的积为常数时,当它们相等时和取得最小值;两个正数的和为常数时,当它们相等时积取得最大值.下面举例说明如何应用均值定理求代数式的最值(最大值或最小值).
The mean theorem of two positive numbers is the compulsory part of high school mathematics, which is often used in the proof of inequalities and the sums of the algebraic formulas, so students are required to master them.First, the mean theorem of two positive numbers means that if a, b ∈ (0, + ∞), then a + b / 2≥ab ~ (1/2), and if and only if a = b Equal sign. The content can generally be summarized as: Arithmetic mean of two positive real ((a + b) / 2) is not less than their geometric mean (ab ~ (1/2)), and secondly, from the mean theorem: when the product of two positive numbers is a constant, When the sum of two positive numbers is a constant, the maximum value is obtained when they are equal. The following gives an example of how to apply the mean theorem to find the maximum value (maximum value or minimum value) of an algebraic equation.