Question

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

• A. maxima at $x=0$
• B. minima at $x=0$
• C. point of inflexion at $x=0$
• D. none of the above

Answer 1

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

Let $S=f^2\left(x\right)+\left(f^{\prime }\right(x){)}^2$
$S^{\prime }=2f\left(x\right).f^{\prime }\left(x\right)+2f^{\prime }\left(x\right)f^{\prime \prime }\left(x\right)=2f^{\prime }\left(x\right)\left[f\right(x)+f^{\prime \prime }(x\left)\right]=-2xg\left(x\right)\left(f^{\prime }\right(x){)}^2$
at $x=0$
For $x\to 0^{-}$
$\Rightarrow x<0\Rightarrow S^{\prime }=-2xg\left(x\right)\left(f^{\prime }\right(x){)}^2>0$
For $x\to 0^{+}$
$\Rightarrow x>0\Rightarrow S^{\prime }=-2xg\left(x\right)\left(f^{\prime }\right(x){)}^2<0$
$S^{\prime }$ changes its sign from positive to negative
$\Rightarrow f^2\left(x\right)+\left(f^{\prime }\right(x){)}^2$ has maxima at $x=0$