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Question:

For function f(x) = xcos(1/x), x ≥ 1, which of the following statements is true?

limx→∞f'(x) = 1

for all x in the interval [1, ∞), f(x+2) - f(x) > 2

for at least one x in interval [1, ∞), f(x+2) - f(x) < 2

f(x) is strictly decreasing in the interval [1, ∞)

Solution:

Given f(x) = xcos(1/x), x ≥ 1 ⇒ f'(x) = cos(1/x) + (1/x)sin(1/x) → 1 for x → ∞. Also f''(x) = -(1/x²)sin(1/x) + (1/x²)(-cos(1/x)) - (1/x³)sin(1/x) = -(1/x²)sin(1/x) - (1/x²)cos(1/x) - (1/x³)sin(1/x) = -(1/x²)sin(1/x) - (1/x²)cos(1/x) - (1/x³)sin(1/x) < 0 for x ≥ 1 ⇒ f'(x) is decreasing for [1, ∞) ⇒ f'(x+2) < f'(x). Also, limx→∞f(x+2) - f(x) = limx→∞[(x+2)cos(1/(x+2)) - xcos(1/x)] = 2 ∴ f(x+2) - f(x) > 2 ∀x ≥ 1