Let  $f:[0,8]\to R$ be twice differentiable function such that  $f\left(0\right)=0,f\left(4\right)=1, f\left(8\right)=1$ then identify the CORRECT statement(s).

A)  Apply LMVT on (0,4) 
 $f^{/}\left(c\right)=\frac{f\left(4\right)-f\left(0\right)}{4-0}=\frac{1}{4}forc\in (0,4)$
B)  Apply LMVT on (0,4) and (4,8)
 $f^{/}\left(c_1\right)=\frac{1}{4}for{c}_1\in (0,4)and{f}^{/}\left(c_2\right)=0for{c}_2\in (4,8)$
$f^{/}\left(x\right)$  is continuous as f(x) is twice differentiable at some point c, the value of  $f^{/}\left(c\right)=1/12$

D)  Consider  $g\left(x\right)=\underset{0}{\overset{x^3}{\int }}f\left(t\right)dt$
 Apply LMVT on (0, 1) and (1, 2)
$$\begin{gathered} g^{\mathbf{|}}\left(\alpha \right)=\frac{g\left(1\right)-g\left(0\right)}{1-0},\alpha \in (0,1) \\ {g}^{\mathbf{|}}\left(\beta \right)=\frac{g\left(2\right)-g\left(1\right)}{2-1},\beta \in (1,2) \\ \Rightarrow g\left(2\right)=g^{\mathbf{|}}\left(\alpha \right)+g^{\mathbf{|}}\left(\beta \right)=3{\alpha }^{2}f\left({\alpha }^3\right)+3{\beta }^{2}f\left({\beta }^3\right) \end{gathered}$$