Ah. What you have here is a ‘poorly posed question’, begging your pardon. If there is a fastest curve, then it seems natural to ask whether there is a slowest curve, but the question doesn’t end up anywhere interesting.
Let’s deal with a similar problem - what’s the shortest curve between two points? You can put a piece of string between the two and pull on it. This is mechanically minimising the length of the string between the two points. The maths effectively does the same thing: it finds the curve where any deviation makes the curve longer, just as it found the slope where any deviation makes the ball take a longer time to roll down it under gravity.
Now what is the longest curve between two points? You don’t find a limiting solution because there is always another curve that can have an extra wiggle in it and be longer.
What are the solutions for the makristochrone? If we only allow solutions where the ball rolls all the way and doesn’t get stuck, then we have to have some gradient at the start. But there will always be a curve with half the gradient at the start. The final solution would be a straight line with an infinitesimal gradient, followed by a sheer drop.
Hooray! A right angle (a flat bit and a sheer drop) is a solution! Well, actually, no it isn’t, really. What we have is a strategy for getting incredibly close to the limiting rule that the ball can’t get stuck. But the solution lies on the limiting rule, and not just close to it, and mathematicians are very particular about that sort of thing. It may seem like a lot of fuss over nothing but the rules are important.
If you were to be making something out of wood, something just short of a right angle would be the solution to all practical purposes. The maths does converge toward a single shape, unlike the ‘longest curve between two points’ which has infinite solutions.