Does anyone know of an easily-describable distribution on $[0,1]$ with a density $p$ (with respect to Lebesgue measure) that satisfies the following properties:

- $p$ is $C^\infty$
- $p(0) = a$, $p(1) = b$ (for fixed real numbers $a$, $b$)
- every derivative of $p$ at 0 and 1 is 0
- p is computable (informally, we can compute $p$ at any point to arbitrary accuracy)