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Best least-square approximation from $p_2$ to a function $f$ on $[0, 1]$ \& Hilbert matrix.

Find $a_0, a_1, a_2 \in R$ s.t.$(\int_0^1 [a_0x^0 + a_1x + a_2x^2 - f(x)]^2 dx)^{\frac{1}{2}} = M$ is the smallest.

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