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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.

Thank you

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Is $X = f_1(x) + f_2(x)$ or is $f_1(x) + f_2(x)$ maximized when $x = X$? Similarly, is $X_1 = f_1(x)$ or is $f_1(x)$ maximized when $x = X_1$? Same question applies to $f_2(x)$. – Jacob Feb 19 '13 at 23:07

$$ \max (f+g) \le \max f + \max g $$

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