The set of values of x for which $f\left(x\right) = 3x^4 – 8x^3 – 6x^2 + 24x – 12$ is an increasing function is

$\begin{array}{rcl}\mathrm{f}\left(\mathrm{x}\right)& =& 3{\mathrm{x}}^4-8{\mathrm{x}}^3-6{\mathrm{x}}^2+24\mathrm{x}-12\\ & \Rightarrow & {\mathrm{f}}^{\mathrm{I}}\left(\mathrm{x}\right)=12{\mathrm{x}}^3-24{\mathrm{x}}^2-12\mathrm{x}+24\\ & =& 12\left({\mathrm{x}}^3-2{\mathrm{x}}^2-\mathrm{x}+2\right)\\ & =& 12\left(\mathrm{x}\left({\mathrm{x}}^2-1\right)-2\left({\mathrm{x}}^2-1\right)\right)\\ & =& 12\left(\mathrm{x}-2\right)\left(\mathrm{x}-1\right)\left(\mathrm{x}+1\right)\end{array}$

$\mathrm{f}\left(\mathrm{x}\right)$ function is monotonically increasing

 $\Rightarrow {\mathrm{f}}^{\mathrm{I}}\left(\mathrm{x}\right)\ge 0$ $\Rightarrow 12\left(\mathrm{x}+1\right)\left(\mathrm{x}-1\right)\left(\mathrm{x}-2\right)>0$

$\mathrm{x}\in \left(-1,1\right)\cup \left(2,\infty \right)$