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Let f: (−1, ∞) → R be defined by f(0) = 1 and f(x) = (1/x) logₑ (1 + x), x ≠ 0. Then the function f:

1. decreases in (−1, 0) and increases in (0, ∞)
2. decreases in (−1, ∞)
3. increases in (−1, ∞)
4. increases in (−1, 0) and decreases in (0, ∞)

Correct answer: decreases in (−1, ∞)

Solution

g(x)=ln(1+x)/x decreases throughout (-1, infinity): e.g. g(-0.9)=2.56, g(-0.5)=1.39, g(0)=1, g(1)=0.69. So f decreases in (-1, infinity).

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