8- Inequalities

A continuous map $f : [0, 1] → [− \dfrac{1}{3} ,  \dfrac{2}{3}]$ is onto and satisfies 

$\int_0^1f(x)dx =0$. 

Prove that 

$\int_0^1f^3(x)dx  \leq \dfrac{1}{9}$.

Proposed by Mihai Piticari, Campulung Moldovenesc, Romania

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