# Maximizing a summation without equals its derivation to zero

1. Jul 20, 2011

### sabbagh80

Hi,
I have sent the following question a couple of days ago. I want to ask my question in a more

general way; Could you please guide me if we have some ways to find the maximum value of a

summation without getting derivation and equals it to zero?

My problem:What is the maximum value of the given summation in terms of $k, l$ and $N$ ?
$$max_{0\leq x \leq k} \sum_{(l_1,l_2)\in A} \frac{N!}{(N-l_1-l_2)!l_1!l_2!} x^{l_1}(k- x)^{l_2}(1-k)^{N-l_1-l_2}$$
where $A=\{l_1,l_2 \in \{0,1,...,N\}|l_1+2l_2=l, l_1+l_2\leq N \}$. Also, $l \in \{0,1,...,2N \}$ and $0<k<1$.

Thanks a lot for your participation.

Last edited: Jul 20, 2011