我的上贴是错的。改为：怎么证明y=x^x 有极值点x=1/e? 请参看Bele的帖子。

下面是Bele的帖子：

This is one of the high school math problems in which Calculus can go much deeper.

Result: For each 0 < x < 1/e, there is a y in the interval (1/e, 1) such that x^x = y^y.

Proof:

Using Calculus, it is easy to verify that function f(x) = x^x decreases from 1 to its minimum f(1/e) as x goes from 0 to 1/e and that f(x) increases from f(1/e) to the positive infinity as x goes from 1/e to the positive infinity.The result then follows.

谢谢！