Let b be a nonzero real number. Suppose f : ℝ → ℝ is a differentiable function such that f 0 = 1 . If the derivative f ' of f satisfies the equation f ' x = f x b 2 + x 2 for all x ∈ ℝ , then which of the following statements is/are TRUE?
C. f x   f - x = 1 for all x ∈ ℝ
f ' x = f x b 2 + x 2 ⇒ f ' x f x = 1 b 2 + x 2 ⇒ l n f x = 1 b tan - 1 x b + C ∴ f 0 = 1         ∴ 0 = 0 + C ⇒ C = 0 ∴   l n f x = 1 b tan - 1 x b ⇒ f x = e 1 b tan - 1 x b ∴ f ' x = e 1 b tan - 1 x b . 1 b 2 + x 2 > 0 for x ∈ R ⇒ f x is increasing for x ∈ R . f x . f - x = e 1 b tan - 1 x b · e - 1 b tan - 1 x b = e 0 = 1 .
Related: Mathematics — Differential Equations · All PYQ Banks