I am looking for a 5 parameter family of analytic functions $f:[0,1]\to \mathbb{R}$ such that

(0) $f$ has zeros at $0,p,1$.

(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, and $f$ depends continuously on these parameters.

A closed form solution in terms of rational operations and elementary functions is preferred. (I know a solution using a cubic spline with 7 knots, but due to its piecewise construction it is nonanalytic.)

I would already be happy with a 4 parameter family where $p$ is determined implicitly or explicitly by the other 4 parameters, since in spite of many attempts, I could not even satisfy this weaker goal.