Putnam 1973 B4

(a) On $[0, 1]$ , let $f(x)$ have a continuous derivative satisfying $0 <f'(x) \leq1$ . Also suppose that $f(0) = 0.$ Prove that $$\left( \int_{0}^{1} f(x) dx \right)^{2} \geq \int_{0}^{1} f(x)^{3} dx.$$ (b) Show an example in which equality occurs.