The derivative of a function is given by f'(x) = (192x³)/(2 + sin⁴(π x)) for all x ∈ R, and it is known that f ((1)/(2)) = 0. If the integral ∫_(1/2)¹ f(x) dx is bounded such that m ≤ ∫_(1/2)¹ f(x) dx ≤ M, what are the possible values of m and M?

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m = 1, M = 12. The integral bounds for f(x) are determined by evaluating the given derivative and considering the constraints. The resulting range for the integral is 1 ≤ ∫ f(x) dx ≤ 12.

The derivative of a function is given by f'(x) = (192x³)/(2 + sin⁴(π x)) for all x ∈ R, and it is known that f ((1)/(2)) = 0. If the integral ∫_(1/2)¹ f(x) dx is bounded such that m ≤ ∫_(1/2)¹ f(x) dx ≤ M, what are the possible values of m and M?

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