Down the slide

As I took this picture of my son on a slide, I was reminded of the Mean Value Theorem for derivatives – that is, the curvy slide is exactly as steep as the straight slide in at least one location (probably three).
But which slide is faster?  What shape would the fastest possible slide have (starting and ending at given points)?  This is called the brachistochrone problem, posed by Johann Bernoulli in 1696.  I gave it a shot once, and didn’t have much luck.  Leave it to Newton to solve that bad boy in one day.  Some day, though, I’ll put a brachistochrone slide in my yard.