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)

$f'(x)=\frac{f(x)}{b^{2}+x^{2}} $
$ \Rightarrow \int \frac{f^{\prime}(x) d x}{f(x)}=\int \frac{d x}{b^{2}+x^{2}} $
$\Rightarrow \ell n|f(x)|=\frac{1}{b} \tan ^{-1}\left(\frac{x}{b}\right)+c$
As $f (0)=1 \Rightarrow \ell n |1|=0+ c \Rightarrow c =0$
$\Rightarrow f ( x )= e ^{\frac{1}{b} \tan ^{-1}\left(\frac{x}{b}\right)}$
$\Rightarrow f ( x ) \cdot f (- x )=1$,
Also $f ( x )$ is increasing $\forall b \in R$.