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In the following equation:$$f(y)=\sup_{x>0}\bigl(\exp(|y|-|y-x|)\bigr)$$ How can I find the value of supremum? can anyone help me to find it?
Thank you.

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up vote 2 down vote accepted

You want to maximize $|y| - |y-x|$ and since the contribution of $|y-x|$ is always $\leq 0$ while that of $|y|$ is always $> 0$ you see that the supremum is attained when $x = y$, and hence corresponds to $\exp(|y|)$.

EDIT: I haven't noticed the condition $x > 0$. As pointed out by Henry for $y < 0$ the supremum is attained when $x \rightarrow 0$ and is $\exp(0) = 1$.

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@ blabler Thank you for your response, but why the supremum is in $x=y$? – golfer Nov 30 '12 at 7:51
Because this is when the contribution from the term $|y-x|$ is minimized – blabler Nov 30 '12 at 7:53
Strictly that is only true when $y \ge 0$. Otherwise the supremum is approached when $x$ approaches $0$, and so corresponds to $\exp(0)=1$. – Henry Nov 30 '12 at 7:55
@blabler Thanks for your answer! – golfer Nov 30 '12 at 7:57
@ Henry : Could you please explain more for $y<0$ ? – golfer Nov 30 '12 at 8:09

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