Thanks for the help. I'm a little confused by your answer; it is probably the result of my not being as expert in math as desirable. You appear to say at first that the maximum entropy distribution is some sort of exponential distribution in which the parameter depends on which part of the domain we are in. But then you write "Eliminating these Lagrange multipliers and writing the density in terms of the quantiles yields ... and then you have what looks to me to be a piecewise uniform distribution. How did we get from an exponential distribution to a uniform distribution?

The notation $1_[x,y]$ means a function whose value is 1 on the interval [x,y] and 0 elsewhere?