I have a simple question. This might be a theorem somewhere, but I do not know the appropriate keywords to find it. Please help.
Say there is a function $G(k,x) = \int_a^x f(k, t) dt$, and I wish to maximize $G(k,x)$ w.r.t. $k$. Under what conditions is this maximization problem equivalent to maximizing $f(k,t)$ w.r.t. $k$?
In short, when is the following true:
$\max_{k} \left\{\int_a^x f(k,t)\right\} dt \Leftrightarrow \max_{k} \{f(k,t)\}$