Let 𝑏 be a nonzero real number. Suppose $$f:R \rightarrow R$$ 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 $$x \in R$$, 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 an decreasing function

C

$$f(𝑥)𝑓(−𝑥) = 1$$ for all $$𝑥 \in R$$

D

$$f(𝑥) -f(−𝑥) = 0$$ for all $$𝑥 \in R$$