Suppose you have a right triangle ABC with hypotenuse AB, AC is along gravity direction, C is the right angle. c1 is AB, c2 is a smooth and convex curve within the triangle connecting A and B. You don't know anything else about the triangle or curve c2.
If it takes a particle without initial speed time t1 to travel c1, which is AB and time t2 to travel curve c2, can you compare t1 and t2? Suppose there is no friction.
Now please note that this is not a classic problem, so the solution is not Brachistochrone curve. You need to figure out what the problem is asking first.
Any feedback would be appreciated.