Question: Where does the function $f(x) = x^{1-x}$ attain its local extrema?
Now analytically one can say that for $x > 0$, we have $x^{1-x} = e^{(1-x)\ln(x)}$, and then we can find the derivative and so forth to find a local extremum at x = 1. But there are more local extrema for $x < 0$ as can be seen from the graph here.
My question is: is there an analytic way to find the other extrema?