Consider the numbers $x^x$,$x^{(x^x)}$,$x^{(x^{(x^x)})}$, etc.
Let $n$ be the number of times $x$ appears in the power tower and $f_n$ the corresponding function, for example $f_4(x)=x^{(x^{(x^x)})}$.
a. It seems that the function $f_n$ has a (probably global) minimum inside the interval $[0,1]$ if and only if $n$ is even. Can this be proven?
b. What can be said about the value of $$I_n=\int_0^1 f_n(x) dx?$$ It seems that the sequence $(I_{2n})$ decreases and the sequence $(I_{2n-1})$ increases.
c. Finally, what about the limit of $I_n$ when $n\to\infty$? Does it exist, and can its value be given exactly or only by numerical calculation?