Let $\mathscr F$ be the set of all continuous functions $f : [1,3] \longrightarrow [-1,1]$ such that $\int_{1}^{3} f(x)\ dx = 0.$ Find the value of $$\sup\limits_{f \in \mathscr F} \int_{1}^{3} \frac {f(x)} {x}\ dx.$$
This is one of the PhD screening test questions I have recently come across which I cannot able to answer. Any help in this regard would be warmly appreciated.
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