# Maximum and minimum of an integral under integral constraints.

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) $\lim_{x \to 0}\frac{x}{f(x)}=0$

• Are we allowed to make addtional hypotheses on $f$? For example, continuity, integrability, etc? Commented Aug 1, 2013 at 20:59
• Yes, you are allowed to assume $f(x)$ as a continuous, differentiable, integrable function. Commented Aug 2, 2013 at 5:07
• 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. Commented Aug 15, 2013 at 11:44
• Are you searching for $p$ or $f$? Which is it? Commented May 16, 2015 at 14:02