I have $f(x)=x^{x}$ and the question,

Which value is larger, $f(1/2)$ or $f(1/\pi)$?

It is easy to find the answer numerically. But, I am keen on an analytical one, with only elementary argument.

  • 5
    $\begingroup$ I'm voting to close this question because it is too broad. There are many posts on this site about that function. Since you have thought about it for a long time you should have answers, or at least attempts, at each of your questions. If you are stuck, ask them one at a time. $\endgroup$ Aug 10 '19 at 15:34
  • 7
    $\begingroup$ It's easier to think of it as $e^{x\log x}$, so for example at zero it extends continuously by taking $1$ as its value, it is decreasing from $0$ to $e^{-1}$ and increasing from there, the range is $[\frac{1}{e^{\frac{1}{e}}}, \infty)$ etc $\endgroup$
    – Conrad
    Aug 10 '19 at 15:34
  • 4
    $\begingroup$ Try graphing it. It's more helpful than you think. $\endgroup$ Aug 10 '19 at 15:35
  • 1
    $\begingroup$ Looking at its graph will be helpful e.g. wolframalpha.com/input/?i=x%5Ex $\endgroup$
    – Story123
    Aug 10 '19 at 16:03

Let $g(x)=\ln f(x)= x \ln x$. Then $g$ is strictly antitone on $(0,1/e)$, so $$g(\tfrac{1}{4})>g(\tfrac{1}{\pi})\text{.}$$ But $$g(\tfrac{1}{4})=g(\tfrac{1}{2})\text{,}$$ so we are done.

  • $\begingroup$ ...... nice ....... $\endgroup$
    – Quanto
    Aug 10 '19 at 17:37

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