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Let f: (0, infinity) -> R be defined by f(x) = integral from 1/x to x of [e^(-(t + 1/t)) / t] dt. Which of the following statements are true? (A) f(x) is monotonically increasing on [1, infinity). (B) f(x) is monotonically decreasing on (0, 1). (C) f(x) + f(1/x) = 0 for all x in (0, infinity). (D) f(2^x) is an odd function of x on R.

1. A and D only
2. A, C and D only
3. B and C only
4. A, B and C only

Correct answer: A, C and D only

Solution

By Leibniz differentiation, f'(x) = e^(-(x+1/x)) * (1/x + 1/x) /... simplifies to positive for x >= 1, confirming (A). Substituting 1/x for x shows f(1/x) = -f(x), confirming (C). From (C), f(2^x) + f(2^(-x)) = 0, making f(2^x) odd in x, confirming (D). For (B), the function is actually increasing (not decreasing) on (0,1), so (B) is false.

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