Question

A function is continuous \& differentiable on $R _0$ and satisfies the condition $x f^{\prime}( x )+f( x )=1$ throughout its domain, with $f(1)=2$. Then the range of the function is

• A. $(-\infty, \infty)$
• B. $(-\infty, 1) \cup(1, \infty)$
• C. $(0, \infty)$
• D. $(1, \infty)$

Answer 1

Answer: (B)

$x f(x)=x+C $
$f (1)=1+ C \Rightarrow C =1 $
$f(x)=\frac{x+1}{x}$
$f ( x )=1+\frac{1}{ x } ; f ^{\prime}( x )=-\frac{1}{ x ^2} $
$\Rightarrow$ fis always derivable and decreasing in its domain $\Rightarrow$ monotonic also $f$ is not bounded.
The graph of $y=f(x)$ is as shown $y=1$ and $x=0$ are the two asymptotes and range is $R -\{1\}$.