Find the positive real number(s) $c$ such that the graphs of $y=x^c$ and $x=y^c$ intersect (somewhere) at an angle $\frac{\pi}{4}$ Find the positive real number(s) $c$ such that the graphs of $y=x^c$ and $x=y^c$ intersect (somewhere) at an angle $\frac{\pi}{4}$.
A problem from Mark Krusemeyer. Should one be using the symmetry of the graphs along $x=y$ due the functions being inverses of one another? Just a thought.

It is possible to use the symmetry as a slight shortcut to Simon's answer: Given the graph and conditions as laid out there, the relevant point of intersection must be at $x=y=1$.  Because of the symmetry, the angles formed by the tangent lines to the curves at that point and the line $y=x$ must be $\frac{\pi}{8}$, so the slope of the tangent line to the blue curve (lower for $0< x<1$) at $x=1$ must be $\tan\frac{3\pi}{8}=1+\sqrt{2}$.  Now, either $cx^{c-1}|_{x=1}=c=1+\sqrt{2}$ or $c^{-1}x^{1/c-1}|_{x=1}=\frac{1}{c}=1+\sqrt{2}$, so $c=1+\sqrt{2}$ or $c=\frac{1}{1+\sqrt{2}}=-1+\sqrt{2}$.
