We provide the best constant $C$ as well as 
all the extremals for the generalized Poincar\'e inequality
    \begin{equation*}
        \int_{0}^{T} a|u|^p\le
        C\int_{0}^{T} a|u^\prime |^{p}
    \end{equation*}
where $a\in L^\infty([0,T])$ satisfies $1 \le
a(t) \le L$, $u\in W^{1,p}_0([0,T],\RR^N)$, $N\ge1$, $p>1$ and $T>0$..