$ max f(\beta)=\frac{\beta}{1+\beta}\cdot \left(1- \frac{\binom{N+B}{B}\cdot\beta^B} {\sum_{i=0}^B {\binom{N+i}{i} \cdot \beta^i}} \right)$
where $\beta\in[0,\infty)$, $N$ and $B$ are identified positive integer (i.e., not variables).
How to obtain the maximum point and the maximum function value in closed form? Or how to prove the maximum point exists and is unique, and give the implicit expression about the maximum point?