Let f: R R be a function such that f(x) + 3f ( 2 - x ) = x, x R . Let the maximum value of f on R be . If the area of the region bounded by the curves g(x) = x^2 and h(x) = x^3, 0, is ^2, then 30 ^3 is equal to _______.

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Let f: R R be a function such that f(x) + 3f ( 2 - x ) = x, x R . Let the maximum value of f on R be . If the area of the region bounded by the curves g(x) = x^2 and h(x) = x^3, > 0, is ^2, then 30 ^3 is equal to _______.

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Correct answer: A. A

Given f(x) + 3f ( 2 - x ) = x Replacing x with 2 - x, we get: f ( 2 - x ) + 3f(x) = ( 2 - x ) = x Multiplying this equation by 3 and subtracting the first equation gives: 9f(x) - f(x) = 3 x - x 8f(x) = 3 x - x f(x) = 3 x - x 8 The maximum value of f(x) is = 3^2 + (-1)^2 8 = 10 8 . Thus, ^2 = 10 64 = 5 32 . The points of intersection of the curves g(x) = x^2 and h(x) = x^3 are given by: x^2 = x^3 x^2(1 - x) = 0 x = 0, x = 1 The area of the region bounded by the curves is: _ 0 ^ 1/ (x^2 - x^3) dx = [ x^3 3 - x^4 4 ]_ 0 ^ 1/ = 1 3 ^3 - 1 4 ^3 = 1 12 ^3 Given that the area is ^2, we have: 1 12 ^3 = 5 32 ^3 = 32 60 = 8 15 Therefore, 30 ^3 = 30 8 15 = 16. Answer: 16

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