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Find the range of the function f(x) = 2*sin²(x) - 3*sin(x) + 8, where f: R -> R.

1. [55/8, 13]
2. [7, 13]
3. [55/8, 8]
4. [6, 13]

Correct answer: [55/8, 13]

Solution

Let t = sin(x), t in [-1, 1]. g(t) = 2t² - 3t + 8. Vertex at t = 3/(2*2) = 3/4, which lies in [-1, 1]. Minimum value g(3/4) = 2*(9/16) - 3*(3/4) + 8 = 9/8 - 9/4 + 8 = 9/8 - 18/8 + 64/8 = 55/8. Maximum at endpoints: g(-1) = 2 + 3 + 8 = 13; g(1) = 2 - 3 + 8 = 7. So max = 13. Range = [55/8, 13].

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