The problem I have is that I have data at two points $x_1,x_2$ and $x_2>x_1>0$. At these two points, I know that the function $f$ has values $f(x_1)$ and $f(x_2)$ respectively.
It is also known that the derivatives of $f$ at these two points i.e., $f'(x_1)$ and $f'(x_2)$.
Now I want to find an interpolating polynomial such that it is convex in the region $x>0$ and also non-negative. If is not possible, at least in the domain $[0,x_2]$.
I tried splines, monotonic cubic splines but they either not guarantee convexity or single interpolating polynomial.