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Find the maximum and minimum of the following integral in terms of $f(x),a,C$:
\begin{align}I=\int_{0}^{a} \frac{x}{f(x)}p(x)dx \end{align} s.t.:
1) $\int_{0}^{a} p(x)dx=1$
2) $\int_{0}^{a} f(x)p(x)dx=C$

Notes:
1) $p(x)$ is an unknown probability density function.
2) $x,f(x),p(x)\geq 0 \;,\; C>0$
3) $f(0)=0$
4) $\lim_{x \to 0}\frac{x}{f(x)}=0$

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  • $\begingroup$ Are we allowed to make addtional hypotheses on $f$? For example, continuity, integrability, etc? $\endgroup$ – TZakrevskiy Aug 1 '13 at 20:59
  • $\begingroup$ Yes, you are allowed to assume $f(x)$ as a continuous, differentiable, integrable function. $\endgroup$ – Amir Kazemi Aug 2 '13 at 5:07
  • $\begingroup$ I have tried some methods in calculus of variations, like Lagrangian multipliers and methods in isoperimetric problems. While I am not sure that I have used those methods correctly, I would like to now if there is any other analytic solution. $\endgroup$ – Amir Kazemi Aug 15 '13 at 11:44
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    $\begingroup$ Are you searching for $p$ or $f$? Which is it? $\endgroup$ – Mercy King May 16 '15 at 14:02

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