TITLE: Largest circle between $y=x^n$ and $y=\sqrt[n]{x}$
QUESTION [19 upvotes]: Something I have been wondering about for a while. Let us look at the area between $x^n$ and $\sqrt[n]{x}$ when $x\in [0,1]$. Where $n$ is a positive integer. 
Below is an image. 
With a given n, how large of a circle can you fit between $x^n$ and $\sqrt[n]x$?
I know that the circle must lie on the line y=x, but more than that I have not been able to figure out. Will update with my research and images. soon =)
Cheers for all help and solutions

REPLY [2 votes]: What is at stake here is an optimization problem with a parameter, namely $n$. In such a case it is easily possible that the "maximum" (or whatever) not only depends numerically on the value of the parameter, but that the extremal situation changes drastically at certain values of the parameter. This is the case in this example: For $n\leq5$ the largest circle touches the two curves $\gamma:\ y=x^n$, $\gamma':\ y=x^{1/n}$ in the two points where $y'(x)=1$; but for $n\geq6$ the  curvature of $\gamma$ and $\gamma'$ in these two points is too large to make this possible. This implies that for $n\geq6$ the extremal situation is globally completely different: The extremal circle will now touch the curves $\gamma$ and $\gamma'$ in totally 4 points. This will make the computing of the extremal circle much more difficult in the case $n\geq6$.