My question is as follows:
Given a function $f: [-h,\infty) \rightarrow \mathcal{R}$ and the maximum function given by
\begin{equation} \max_{s\in [-h,0]} |f(t+s)| \end{equation} for $t\geq0$.
Then I would like to know the absolute value of the derivative with respect $t$ of the above function or at least an upper bound. That is,
\begin{equation} \left| \frac{d}{dt}\left( \max_{s\in [-h,0]} |f(t+s)| \right) \right| \end{equation}
My hypothesis is that it can be bound as follows
\begin{equation} \left| \frac{d}{dt}\left( \max_{s\in [-h,0]} |f(t+s)| \right) \right| \leq \max_{s\in [-h,0]} \left( \frac{d}{dt}|f(t+s)| \right) \end{equation}
The idea is that the maximum function can not change faster than the function at the maximum point, but I struggle to formally prove it.
Can anyone suggest me how to prove it? or if there is already a proof.
Thank you.