Find the maximum value of a summation

[sabbagh80]

Hi,
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 $A=\{(l_1,l_2)|l_1,l_2 \in \{0,1,2,...,N\}$ and $l_1+2l_2=l\}$ and $0<k<1$.
Thanks a lot for your participation.

 

[hunt_mat]

Forgetting about all the constants for a moment, you have a function $x^{a}(k-x)^{b}$, the way to look at this is differentiate it ans et it to zero, do this for the simple function and see what you get.

 

[sabbagh80]

Forgetting about all the constants for a moment, you have a function $x^{a}(k-x)^{b}$, the way to look at this is differentiate it ans et it to zero, do this for the simple function and see what you get.

I had done it before. it is $(\frac{k l_1}{l_1+l_2})^{l_1} (\frac{k l_2}{l_1+l_2})^{l_2}$.but the problem is that each term of the summation is maximized in different values of the given interval as $l_1, l_2$ vary.