Problem: Given $f:[0,1] \to \mathbb{R}$ is integrable over $[0,1]$, and that $$\int_0^1 dx \, f(x) = \int_0^1 dx \, x f(x) = 1$$ show that $$\int_0^1 dx \, f(x)^2 \ge 4$$ The way to the solution here is not trivial. I started by always recognizing that, with integral inequalities, it never hurts to start with […]

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