Let b be a nonzero real number and f: R - R be differentiable with f(0) = 1. If f'(x) = f(x)/(b² + x²) for all real x, which of the following are TRUE? (A) If b 0, then f is increasing. (B) If b

Question

Let b be a nonzero real number and f: R -> R be differentiable with f(0) = 1. If f'(x) = f(x)/(b² + x²) for all real x, which of the following are TRUE? (A) If b > 0, then f is increasing. (B) If b < 0, then f is decreasing. (C) f(x)*f(-x) = 1 for all x. (D) f(x) - f(-x) = 0 for all x.

Accepted Answer

(A) and (C). Since b² + x² > 0 and f > 0, f'(x) > 0 so f is always increasing regardless of sign of b; thus (A) is true and (B) false. Also d/dx[f(x)f(-x)] = 0 with value 1 at x=0, so (C) is true; (D) is false.

Let b be a nonzero real number and f: R -> R be differentiable with f(0) = 1. If f'(x) = f(x)/(b² + x²) for all real x, which of the following are TRUE? (A) If b > 0, then f is increasing. (B) If b < 0, then f is decreasing. (C) f(x)*f(-x) = 1 for all x. (D) f(x) - f(-x) = 0 for all x.

Solution

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