Question

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

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

Answer 1

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

$h\left(x\right)=f^2\left(x\right)+{\left\{f^{\prime }\right(x\left)\right\}}^2{h}^{\prime }\left(x\right)=2f\left(x\right)f^{\prime }\left(x\right)+2f^{\prime }\left(x\right)f^{\prime \prime }\left(x\right)=2f^{\prime }\left(x\right)\left(f\right(x)+f^{\prime \prime }(x\left)\right)=2f^{\prime }\left(x\right)[-xg(x\left)f^{\prime }\right(x\left)\right]=-2xg\left(x\right){\left\{f^{\prime }\right(x\left)\right\}}^2{h}^{\prime }\left(0^{+}\right)>0and{h}^{\prime }\left(0^{-}\right)<0and{h}^{\prime }\left(0\right)=0h^{\prime }\left(0^{+}\right)>0and{h}^{\prime }\left(0^{-}\right)>0$

$\Rightarrow$ function $h\left(x\right)$ must be something like this

Hence, $f^2\left(x\right)+\left(f^{\prime }\right(x){)}^2$ has a maxima at $x=0$