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$ 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?

JLiu
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  • Can you not find where the derivative vanishes? – AnonSubmitter85 Apr 14 '14 at 07:56
  • You could try to simplify first, e.g. since $N$ and $B$ are identified positive integers, then ${N+B\choose B}$ can be replaced by a constant, $\lambda$, say. You may also be able to simplify the sum in the denominator using the Binomial Theorem somehow. – pshmath0 Apr 14 '14 at 07:57
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    @pbs Also, the denominator looks like it might simplify to some sort of binomial. – AnonSubmitter85 Apr 14 '14 at 07:59
  • @AnonSubmitter85 thank you very much for your comments, but the denominator is the sum of $\binom{N+i}{i}$, not the $\binom{N}{i}$, so I cannot simplify it by now, so cloud you help me? I can derive the equation that the first-order derivative equals to zero, but it should be verified that this point is maximum point. I really appreciate you and expect your help. – JLiu Apr 14 '14 at 08:29
  • @pbs thank you very much for your comments, but the denominator is the sum of (N+ii), not the (Ni), so I cannot simplify it by now, so cloud you help me? I can derive the equation that the first-order derivative equals to zero, but it should be verified that this point is maximum point. I really appreciate you and expect your help. – JLiu Apr 14 '14 at 08:32
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    @user143002 I don't have time right now, but will take a look later unless someone else does. One tip: let $B=0$ and try to solve. Then let $B=1$ and try to solve, and so on. You may see a pattern arise ! – pshmath0 Apr 14 '14 at 08:41
  • @pbs Thank you very much, I have tried this method, but not success, if you are convenient, can you help me? I think maybe let the first-order derivative be 0, and verify that before this point, the first-order derivative is larger than 0, and after this point is less than 0, then this point is the optimal point? – JLiu Apr 15 '14 at 06:12
  • @user14002 Do you know about www.coursera.org ? If you search for the "Calculus One" course there is a whole video section about "Optimization" which will help you find your maximum. – pshmath0 Apr 15 '14 at 07:39

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