For a maximization problem I am required to find and solve an Euler equation. I checked my stepts multiple times but I get stuck on the following and I cannot determine whether it is true or not.
$\frac{\partial}{\partial t}\int_t^T 1-q(s)ds = 0$.
Where $t \in [0,T]$, q:[0,T] $\rightarrow$ [0,1] is a strictly increasing C2 function with q(0) = 0 and q(T) = 1.
Could somebody tell me if this equation holds or that I must have made a mistake earlier?
$\frac{\partial}{\partial t}\int_t^T (1-q(s))ds=-\frac{\partial}{\partial t}\int_T^t (1-q(s))ds=-(1-q(t))=q(t)-1.$
If $\frac{\partial}{\partial t}\int_t^T (1-q(s))ds=0$, then $q(t)=1$ for all $t \in [0,T].$