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Find the value of the constant $$a$$ for which the function $$f ( x ) = a x ^ { 2 }$$ best approximates the function $$g ( x ) = x ^ { 3 }$$ on the interval $$[ 0,1 ]$$ , in the sense that the integral $$I = \int _ { 0 } ^ { 1 } ( f ( x ) - g ( x ) ) ^ { 2 } d x$$
$$a = 5 / 6 , I = 1 / 252$$