Suppose we have to choose a finite or an infinite sequence $\{\lambda_1,\lambda_2,\cdots,\lambda_T\}$ to maximize (or mimizize) $$\sum_{i=1}^{T-1}(\lambda_{i+1}-\lambda_{i})\lambda_{i+1}.$$ When there are restrictions that it should be $\lambda_1=1$ and $\lambda_{T}=0$, how can I proceed and find a solution? $\lambda_i$ should lie in between $0$ and $1$, i.e., $\lambda_i\in[0,1],\forall i$.
If the sequence is infinite ($T=\infty$), we can ignore the restriction $\lambda_T=0$.
In this case, what should be the maximum and minimum (or supremum and infimum) of $\sum_{i=1}^{T-1}(\lambda_{i+1}-\lambda_{i})\lambda_{i+1}$?