$$P(t):=\frac{A^{2}}{N} \sum_{j=1}^{N} \sum_{k=1}^{N} x_{k} x_{j} \exp \left(2 \pi i \frac{B}{N}(k-j) t\right)$$ Could any one tell me how to maximize $P(t)$? $t\in [0,T)$.
I have done the derivative of $P(t)$ and found that $P(t)=0$ iff $j=k$ at that point let's say $t=t_0$, But after that, I am not able to draw any conclusion. Thank you so much for any help. I was given a hint to use $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}$$ and $\sum_j\sum_k x_jx_k=(x_1+\dots+x_N)^2$, $x_j=0$ or $1$, at some stage, but I am not able to realize that either.