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Let f: [1/2, 1] → R (the set of all real numbers) be a positive, non-constant and differentiable function such that f'(x) < 2 f(x) and f(1/2) = 1. Then the value of ∫[1/2 to 1] f(x) dx lies in the interval -

1. (A) (2e - 1, 2e)
2. (B) (e - 1, 2e - 1)
3. (C) (e - 1/2, e - 1)
4. (D) (0, e - 1/2)

Correct answer: (D) (0, e - 1/2)

Solution

The derivative condition f'(x) < 2f(x) implies exponential growth constraints on f(x). Using f(1/2) = 1 and integrating f(x) over [1/2, 1], the result lies in the interval (0, e - 1/2).

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