$$\begin{gathered} \text{ Let }F:R\to R\text{ be a thrice differentiable function. Suppose that }F\left(1\right)=0,F\left(3\right)=-4 \\ \text{ and }{F}^{\mathrm{\prime }}\left(x\right)<0\text{ for all }x\in (1/2,3).\text{ Let }f\left(x\right)=xF\left(x\right)\text{ for all }x\in R \end{gathered}$$

The correct statement is 

• A

  $f^{\mathrm{\prime }}\left(1\right)<0$

• B

  $f\left(2\right)>0$

• C

  $f^{\mathrm{\prime }}\left(x\right)\ne 0\text{ for any }x\in (1,3)$

• D

  $f^{\mathrm{\prime }}\left(x\right)>0\text{ for some }x\in (1,3)$