Is there any relationship similar to the following. Let $X$ be the maximum of functions $f_1(x)+f_2(x)$. Let $X_1$ be a maximum of $f_1(x)$ and let $X_2$ be a maximum of $f_2(x)$. Is there any relationship between $X$ and $X_1$ and $X_2$?
For example can we say under what condition will $X$ be in-between $X_1$ and $X_2$, $X_1 \le X \le X_2$.
Any reference would be greatly appreciated.