Brain Teaser - Minimum distance between two curves

[quinyu]

1. The coordinates at f(x) are (x,x^2+1); at g(y) are (y,sqrt(y)). Using a global minimization for the distance, the lowest distance possible is ~0.427592 where x=~0.331695 and y=~0.568071.
2. This x is the (real) root of the equation 4*x^5+6*x^3-x^2=1/8
3. More precisely y is ~0.5680712803530178751565860996, but I could not find a single proper closed form for this.

Done with Mathematica/WolframAlpha.