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 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

$$\begin{gathered} f\text{'}\left(\alpha \right)g\left(\alpha \right)-g\text{'}\left(\alpha \right)f\left(\alpha \right)={\alpha }^4+2{\alpha }^2+10------\left(1\right) \\ f\text{'}\left(\beta \right)g\left(\beta \right)-g\text{'}\left(\beta \right)f\left(\beta \right)={\beta }^4+2{\beta }^2+10------\left(2\right) \end{gathered}$$

$f\text{'}\left(\alpha \right)<0$ 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

$$\begin{gathered} f\text{'}\text{'}\left(x\right)g\left(x\right)+f\text{'}\left(x\right)g\text{'}\left(x\right)-g\text{'}\left(x\right)f\text{'}\left(x\right)-g\text{'}\text{'}\left(x\right)f\left(x\right)=4x^3+4x \\ f\text{'}\text{'}\left(x\right)g\left(x\right)-g\text{'}\text{'}\left(x\right)f\left(x\right)=4x^3+4x \end{gathered}$$

Substitute $x=\alpha$ and $\beta$

$$\begin{gathered} f\text{'}\text{'}\left(\alpha \right)g\left(\alpha \right)-g\text{'}\text{'}\left(\alpha \right)f\left(\alpha \right)=4{\alpha }^3+4\alpha ----\left(3\right) \\ f\text{'}\text{'}\left(\beta \right)g\left(\beta \right)-g\text{'}\text{'}\left(\beta \right)f\left(\beta \right)=4{\beta }^3+4\beta -----\left(4\right) \end{gathered}$$

From equation 3 and 4,  $\alpha <0 and \beta >0.$