You are right, of course and it solves the problem, as presented.

I read it wrong but the fastest journey from A to B (i.e. displacement) is a more likely scenario to deal with than how quickly you can travel a certain distance.

I guess, if you had to drop a ball down a fixed length of tubing from a given height above ground, the original question would deal with that scenario.

Yes. Note (but probably you already know it) that if, instead, you have to find the quicker path between *two fixed points* A and B with the second at lower height than the first, it would be a different problem, known as the 'Brachistochrone' problem:

http://en.wikipedia.org/wiki/Brachistochrone_curvewhich solution is

~~an hyperbolic cosine~~ a cycloid.

To solve this last problem we can use variation calculus and the resulting Lagrange equation. Only one function y(x) solve the equation (the cycloid).

I say this because I tried to use the same method for the problem I posted and I found that there is no solution to the resulting equation, that is no function h(s) for which the Functional F(h): h(s)-->T (T = total time) is stationary.

For a normal function f(x), if it's never stationary, you can say that its maximum and minimum values (if they exist) must be at the border of the domain. But which is the border of the domain for the functional F(h), if we can use the same reasoning (not sure of it)?

Intuitively it seems that the function h(s) for which the functional is minimum is that one which graph is horizontal for a lenght (π/2 - 1)R and then vertical for a lenght R (kind of opposite of the quicker path). Does it mean that those two functions h(s) represent the 'border' of the domain of F(h)?