When answering [this](https://math.stackexchange.com/questions/116112/find-the-maximum-of-fx-x1-x) question, I contemplated across the following problem:

> 
> Find $x$ such that $x^{\frac{1}{x}}=0.$
> 
> 
> 

I thought RHS $ = 0$ is a special case, so I attempted solving $$\frac{1}{x} \ln x = \ln c$$ from some constant $c > 0,$ but I do not know how. I heard about Lambert W, but I am not familiar with it.

BTW, Maple `solve(x^(1/x) = 0)` gives $x=0$, and [WolframAlpha](http://www.wolframalpha.com/input/?i=solve+x%5E%281%2Fx%29+%3D%3D+0) gives `(no solution exist)`!

How can I solve this?