Merging two univariate functions gracefully

Question

Suppose I tell you that $$ f(0) = 0 $$ $$ f'(0) = 0 $$ and $$ f''(0) = a $$ for known $a>0$, whereas for large $x$ $$ f'(x) \approx \cosh^{-1}(x) $$ for $x>2$. Knowing nothing else about $f(x)$, what is a sensible interpolation consistent with this information?