Question

Let $b$ be a nonzero real number. Suppose $f : R \rightarrow R$ is a differentiable function such that $f(0)=1$. If the derivative $f^{\prime}$ of $f$ satisfies the equation
$f^{\prime}(x)=\frac{f(x)}{b^{2}+x^{2}}$
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 a decreasing function
• C. $f ( x ) f (- x )=1$ for all $x \in R$
• D. $f(x)-f(-x)=0$ for all $x \in R$

Answer 1

Answer: (A), (C)

If $b>0$, then $f$ is an increasing function

Answer 2

(A) If $b>0$, then $f$ is an increasing function
(C) $f ( x ) f (- x )=1$ for all $x \in R$