In the given problem, how can I optimize $f(t)$. Note that this is a problem when we have $n$ objects doing some task and the probability of success of that task is $f(t)$.


\begin{align}
P(success \ of \ a\ task) = \sum^{n-1}_{i=1} t_i\left(1-\dfrac{t_i}{x}\right)\left(1-\dfrac{t_i}{bx}\right)^{n-2} .\ .\ .\ .\ .\ .\ .\ (1)
\end{align}


I tried to do this using the simple optimization problem (I have asked a part of this question earlier as well, [here]). So using differentiation method, I am able to find the optimized f(t) when considering success for only single device


\begin{equation*}
g'(t) = \frac{d}{dt}t\left(1-\dfrac{t}{x}\right)\left(1-\dfrac{t}{bx}\right)^{n-2} = 0
\end{equation*}


\begin{equation*}
t^{**} = -\dfrac{m\sqrt{4n^2-4n+d^2-2d+1}-2mn+\left(1-d\right)m}{2d}
\end{equation*}


But for the case as in equation (1), every $d_{th} \in D$ object can have different value of $t$. Hence optimization seems very difficult.