I am looking for a 5 parameter family of analytic functions $f:[0,1]\to R$ with given zeros at $0,p,1$ such that

(1) $f$ is convex in $[0,p]$ and concave in $[p,1]$.

(2) The five parameters, $p$ with $0<p<1$ and arbitrary positive values of $-f'(0)$, $-f'(1)$, $f''(0)$, and $-f''(1)$, can be independently prescribed.

A closed form solution in terms of rational operations and elementary functions is preferred.