Let $f\left(x\right)$  be a continuous function in $x\in [0,5]$ and twice differentiable function in $x\in (0,5)$ such that $f\left(4\right)=f\left(5\right)=0$ then which of the following is/are correct

Let  $g\left(x\right)=x^{n}f\left(x\right)(g\left(4\right)=g\left(0\right)=g\left(5\right)=0)$
$g^{\text{'}}\left(x\right)=x^n{f}^{\text{'}}\left(x\right)+f\left(x\right)n.x^{n-1}$ 
 $g^{\text{'}}\left(\alpha \right)=0\Rightarrow nf\left(\alpha \right)+\alpha f^{\text{'}}\left(\alpha \right)=0$
BY LMVT on [0,4] & [4,5]
$f^{\text{'}}\left(C_1\right)=\frac{f\left(4\right)-f\left(0\right)}{4}$ , $f^{\text{'}\text{'}}\left(C_2\right)=\frac{f\left(5\right)-f\left(4\right)}{1}$
         = $-\frac{f\left(0\right)}{4}$                   , $f^{\text{'}}\left(C_2\right)=0$ 
 $4f^{\text{'}}\left(C_1\right)=-f\left(0\right)$
 $4f^{\text{'}}\left(C_1\right)+f\left(0\right)+f^{\text{'}}\left(C_2\right)$