Question

A function f (x) is given by the equation , $x^2{f}^{\prime }\left(x\right)+2xf\left(x\right)-x+1=0,f\left(1\right)=0$

$f\left(x\right)$ is

• A. increasing in $(-\infty ,0)\cup (1,\infty )$
• B. decreasing in ( -1, 1 )
• C. non monotonic in ( 0 , 1 )
• D. non monotonic in $(-\infty ,0)\cup (1,\infty )$

Answer 1

The correct option is D non monotonic in $(-\infty ,0)\cup (1,\infty )$

$\frac{d}{dx}\left[x^{2}y\right]=x-1$ where $y=f\left(x\right)$

$x^{2}y=\int (x-1)dx$

This gives $y=\frac{1}{2}-\frac{1}{x}+\frac{1}{2x^2}$