Find all functions $f$ satisgying the $3$ definite integrals

Hölder's inequality implies that for functions $g(x)$ and $h(x)$, $$ \left(\int_0^1 g(x)h(x)\,dx\right)^2\leq \int_0^1 g(x)^2dx \cdot \int_0^1 h(x)^2 dx, $$ with equality if and only if $g(x)$ is a scalar multiple of $h(x)$.

Hint: In your situation, take $g(x)=\sqrt{f(x)}$ and $h(x)=x\sqrt{f(x)}$.

Tags:

Calculus

Definite Integrals

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