An integral inequality

Let f:[0,1] \rightarrow \mathbb{R} be a continuous function such that

(1)   \begin{equation*} \int_0^1 f(x) \, {\rm d}x = \int_0^1 x f(x) \, {\rm d}x =1 \end{equation*}

Prove that \displaystyle \int_0^1 f^2 (x) \, {\rm d}x \geq 4 .

Solution

We note that f(x)=6x-2 satisfies all conditions. Thus:

    \begin{align*} 0 &\leq \int_{0}^{1} \left ( f(x) - 6x+2 \right )^2 \, {\rm d}x \\ &=\int_{0}^{1} f^2 (x) \, {\rm d}x - 4 \end{align*}

and thus the conclusion.

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