I am currently facing problem to solve the following question. Please help!
Let $F = \{f: [1, 3] \rightarrow [-1, 1] \mid f \text{ is continuous and } \int_{1}^{3}f(x)dx = 0\}.$
Find $\sup_{f \in F}\int_{1}^{3}\frac{f(x)}{x}dx$.
I think the supremum will be $\ln(3)$, but I don't know if this is right.