TITLE: Behavior of the Nth root of N?
QUESTION [6 upvotes]: Taking the Nth root of some real number $N$ (ie: $R(N) = N^{1/N}$), generally $R(X) > R(Y)$ when $X < Y$. This obviously isn't the case though when $ X< Y < 3$. Put another way, starting at $N = 0.1$ and increasing $N$ by $0.1$ if we plot each R(N) we see that the result increases and peaks just before it reaches $3$ and then steadily decreases. Can anyone shed some light on this behavior?

REPLY [4 votes]: Write $x^{1/x} = \exp (\frac1x \log x)$. Since the exponential function is monotonic (increasing), it is sufficient to look at the behaviour of $f(x) = \frac1x \log x$. Its derivative is
$$f'(x) = \frac{1 - \log x}{x^2}$$
and that is positive for $\log x < 1 \iff x < e$, and negative for $x > e$, hence it has a (unique) maximum at $x = e$, and therefore $x^{1/x}$ also has a (unique) maximum at $x = e$.