Question

If $f:R\to R$ and $g:R\to R$ are two functions such that $f\left(x\right)+f\text{'}\text{'}\left(x\right)=-xg\left(x\right)f\text{'}\left(x\right)$ and $g\left(x\right)>0\forall x\in R$, then the function $f^2\left(x\right)+\left(f\text{'}\right(x){)}^2$ has

• A. a maxima at $x=0$
• B. a minima at $x=0$
• C. a point of inflexion at $x=0$
• D. None of these

Answer 1

The correct option is A a maxima at $x=0$

$f\left(x\right)+f^{\prime \prime }\left(x\right)=-xg\left(x\right)f^{\prime }\left(x\right)$
Let $h\left(x\right)=f^2\left(x\right)+\left(f^{\prime }\right(x){)}^2$
$\therefore h^{\prime }\left(x\right)=2f\left(x\right)f^{\prime }\left(x\right)+2f^{\prime }\left(x\right)f^{\prime \prime }\left(x\right)$
$=2f^{\prime }\left(x\right)(-x)g\left(x\right)f^{\prime }\left(x\right)$
$=-2x\left(f^{\prime }\right(x\left){)}^{2}g\right(x)$
As,
$h^{\prime }\left(x\right)<0x>0h^{\prime }\left(x\right)>0x<0$
So, $x=0$ is a point of maxima for $h\left(x\right)$.