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.
How can I find the supremum value of this equation?
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calculus
linear-algebra
1 Answers
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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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0That's right, for y < 0 the supremum is 1, I haven't noticed the condition x > 0 – 2012-11-30