Question

Let $b$ be a nonzero real number. Suppose $f:R\to R$ 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)

$f^{\prime }(x)=\frac{f(x)}{b^2+x^2}$
$\Rightarrow \int \frac{f^{\prime }(x)dx}{f(x)}=\int \frac{dx}{b^2+x^2}$
$\Rightarrow \ell n|f(x)|=\frac{1}{b}{\tan }^{-1}\left(\frac{x}{b}\right)+c$
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 $\forall b\in R$.