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Maximize Integral Increasing Functions

2175ExpertCalculusOptimization

64th Romanian Mathematical Olympiad - Final Round

Consider $a \in (0,1)$ and $C$ the set of all increasing functions $f: [0,1] \to [0, \infty)$ such that $\int_0^1 f(x) \, dx = 1$. Determine: a. $\max_{f \in C} \int_0^a f(x) \, dx$, and b. $\max_{f \in C} \int_0^a (f(x))^2 \, dx$.
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