1952 AMC 12 第 45 题

Question

若

a

与

b

是两个不相等的正数，则：

If

a

and

b

are two unequal positive numbers, then:

\dfrac{2ab}{a+b}\gt\sqrt{ab}\gt\dfrac{a+b}{2}

\sqrt{ab}\gt\dfrac{2ab}{a+b}\gt\dfrac{a+b}{2}

\dfrac{2ab}{a+b}\gt\dfrac{a+b}{2}\gt\sqrt{ab}

\dfrac{a+b}{2}\gt\dfrac{2ab}{a+b}\gt\sqrt{ab}

\dfrac{a+b}{2}\gt\sqrt{ab}\gt\dfrac{2ab}{a+b}

识别

a

与

b

的算术平均数、几何平均数和调和平均数

Recognize the arithmetic, geometric, and harmonic means of

a

and

b

对于不相等的正数，各平均值之间的不等式为严格不等式

For unequal positive numbers, the mean inequalities are strict

对不相等的正数

a,b

，算术平均值与几何平均值不等式给出

\frac{a+b}{2}\gt\sqrt{ab}\text{。}

对

\frac{1}{a}

与

\frac{1}{b}

使用同一不等式，得到

\sqrt{ab}\gt\frac{2ab}{a+b}\text{。}

因此，从大到小依次为算术平均数、几何平均数、调和平均数。

因此，正确答案是 E。

The arithmetic-geometric mean inequality gives

\frac{a+b}{2}\gt\sqrt{ab}

for unequal positive

a,b.

Applying the same inequality to

\frac{1}{a}

and

\frac{1}{b}

gives

\sqrt{ab}\gt\frac{2ab}{a+b}.

Therefore the decreasing order is arithmetic mean, geometric mean, harmonic mean.

Thus, the correct answer is E.

其他年份的第 45 题