If  $y=f\left(x\right)$  be continuous concave upward function and  $y=g\left(x\right)$  be a function such that  $f^{\text{'}}\left(x\right)g\left(x\right)-g^{\text{'}}\left(x\right)f\left(x\right)=x^4+2x^2+10$  then which of the following options is (are) CORRECT?

$f\left(x\right)$  is concave upward function, so  $f\text{'}\text{'}\left(x\right)>0$
$f\text{'}\left(x\right)$  is strictly increasing function
$\alpha$  and  $\beta$  be consecutive roots of  $f\left(x\right)(\alpha <\beta )$
 $f\left(\alpha \right)=f\left(\beta \right)=0$
put   $x=\alpha$  and  $\beta$  in the given equation
$f\text{'}\left(\alpha \right)g\left(\alpha \right)-g\text{'}\left(\alpha \right)f\left(\alpha \right)={\alpha }^4+2{\alpha }^2+10------\left(1\right)$
 $f\text{'}\left(\beta \right)g\left(\beta \right)-g\text{'}\left(\beta \right)f\left(\beta \right)={\beta }^4+2{\beta }^2+10------\left(2\right)$
$f\text{'}\left(\alpha \right)<0$  and  $f\text{'}\left(\beta \right)>0$ ( $f\left(x\right)$  is concave upward function)
So  $g\left(\alpha \right)<0$  and  $g\left(\beta \right)>0$  from equation (1) and (2)
Differentiating the given equation, we get
 $f\text{'}\text{'}\left(x\right)g\left(x\right)+f\text{'}\left(x\right)g\text{'}\left(x\right)-g\text{'}\left(x\right)f\text{'}\left(x\right)-g\text{'}\text{'}\left(x\right)f\left(x\right)=4x^3+4x$
 $f\text{'}\text{'}\left(x\right)g\left(x\right)-g\text{'}\text{'}\left(x\right)f\left(x\right)=4x^3+4x$
Substitute  $x=\alpha$  and  $\beta$
 $$\begin{gathered} f\text{'}\text{'}\left(\alpha \right)g\left(\alpha \right)-g\text{'}\text{'}\left(\alpha \right)f\left(\alpha \right)=4{\alpha }^3+4\alpha ----\left(3\right) \\ \Rightarrow f\text{'}\text{'}\left(\alpha \right)g\left(\alpha \right)=\alpha (4{\alpha }^2+4) \\ f\text{'}\text{'}\left(\beta \right)g\left(\beta \right)-g\text{'}\text{'}\left(\beta \right)f\left(\beta \right)=4{\beta }^3+4\beta -----\left(4\right) \\ f\text{'}\text{'}\left(\beta \right)g\left(\beta \right)=\beta (4{\beta }^2+4) \end{gathered}$$
From equation (3) and (4),  $\alpha <0$  and  $\beta >0$
 $f\text{'}\left(x\right)g\left(x\right)-g\text{'}\left(x\right)f\left(x\right)=x^4+2x^2+10$
let be the root between  $\alpha$  and  $\beta$  for g (x)
 $$\begin{gathered} f\text{'}\left(\gamma \right)g\left(\gamma \right)-g\text{'}\left(\gamma \right)f\left(\gamma \right)={\gamma }^4+2{\gamma }^2+10 \\ -g\text{'}\left(\gamma \right)f\left(\gamma \right)={\gamma }^4+2{\gamma }^2+10 \\ f\text{'}\text{'}\left(x\right)>0 \\ f\left(\gamma \right)<0\Rightarrow g\text{'}\left(\gamma \right)>0 \end{gathered}$$
g (x) has exactly one root between  $\alpha$  and  $\beta$