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Consider the following three statements for the function f : (0,∞) → R defined by f(x) = |log_e x| - |x - 1|. (I) f is differentiable for all x > 0. (II) f is increasing in (0,1). (III) f is decreasing in (1,∞). Then.

1. All (I), (II) and (III) are TRUE.
2. Only (I) is TRUE.
3. Only (II) and (III) are TRUE.
4. Only (I) and (III) are TRUE.

Correct answer: Only (I) and (III) are TRUE.

Solution

Statement (I) is true because the function is composed of differentiable components over its domain. Statement (III) is also true as the derivative of the function indicates it decreases for x > 1. However, statement (II) is false since the function is not increasing in the interval (0,1).

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