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)

$xf(x)=x+C$
$f(1)=1+C\Rightarrow C=1$
$f(x)=\frac{x+1}{x}$
$f(x)=1+\frac{1}{x};f^{\prime }(x)=-\frac{1}{x^2}$
$\Rightarrow$ 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 $R-\{1\}$.