Let $F\mathbb{ }:\mathbb{ }\mathbb{R}\to \mathbb{R}$ be a thrice differentiable function. Suppose that $F\left(1\right)=0, F\left(3\right)= -4$ and $F\prime \left(x\right)<0$ all $x\in \left(\frac{1}{2}, 3\right).$ Let $f\left(x\right)=xF\left(x\right)$ for all $x\in \mathbb{ }\mathbb{R}$ .
If $\underset{1}{\overset{3}{\int }}{x}^2{F}^{\prime }\left(x\right)dx= -12$ and $\underset{1}{\overset{3}{\int }}{x}^3 F^{\prime \prime }\left(x\right)dx=40,$ then the correct expression(s) is(are)

• A

  $9f^{\prime }\left(3\right)+f^{\prime }\left(1\right)-32=0$
• B

  $\underset{1}{\overset{3}{\int }}f\left(x\right)dx=12$
• Correct answer
  C

  $9f^{\prime }\left(3\right)-f^{\prime }\left(1\right)+32=0$
• Correct answer
  D

  $\underset{1}{\overset{3}{\int }}f\left(x\right)dx= -12$

Hence, the correct answer is options: 'C', 'D'.