# Which value is larger, $f(1/2)$ vs $f(1/\pi)$, given $f(x)=x^x$

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.

• 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. – Ethan Bolker Aug 10 at 15:34
• 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 – Conrad Aug 10 at 15:34
• Try graphing it. It's more helpful than you think. – infinite-blank- Aug 10 at 15:35
• Looking at its graph will be helpful e.g. wolframalpha.com/input/?i=x%5Ex – Story123 Aug 10 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.