What is the continuous probability distribution that maximizes entropy, given only the bounds of the random variable [a,b] and the mean mu of the probability distribution?

For example:

if a=0, b=1, and mu=0.5, it should return a U[0,1].

if a=10, b=20, and mu=20, it should return Dirac delta at x=20.

if a=0, b=1, and mu=0.8 it should return ... ?

I imagine the general solution will be based on the Beta distribution with some alpha and beta parameters expressed in terms of a, b, and mu, but I don't know.

Many thanks!