Let $b$ be a nonzero real number. Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function such that $f(0) = 1$. If the derivative $f'$ of $f$ satisfies the equation $$f'(x) = \frac{f(x)}{b^{2} + x^{2}}$$ for all $x \in \mathbb{R}$, then which of the following statements is/are TRUE? (A) If $b > 0$, then $f$ is an increasing function (B) If $b < 0$, then $f$ is a decreasing function (C) $f(x)f(-x) = 1$ for all $x \in \mathbb{R}$ (D) $f(x) - f(-x) = 0$ for all $x \in \mathbb{R}$
Let $b$ be a nonzero real number. Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function such that $f(0) = 1$. If the derivative $f'$ of $f$ satisfies the equation
$$f'(x) = \frac{f(x)}{b^{2} + x^{2}}$$
for all $x \in \mathbb{R}$, then which of the following statements is/are TRUE?
(A) If $b > 0$, then $f$ is an increasing function
(B) If $b < 0$, then $f$ is a decreasing function
(C) $f(x)f(-x) = 1$ for all $x \in \mathbb{R}$
(D) $f(x) - f(-x) = 0$ for all $x \in \mathbb{R}$