$\mathit{f}\left(\mathit{x}\right)={\mathit{x}}^3-3{\mathit{x}}^2+12\mathit{x}-18$
$\mathit{f}\prime \left(\mathit{x}\right)=3{\mathit{x}}^2-6\mathit{x}+12=3\left({\mathit{x}}^2-2\mathit{x}+4\right)$
Case I: f is strictly increasing (f'(x) > 0)
$3\left({\mathit{x}}^2-2\mathit{x}+4\right)>0\to {\mathit{x}}^2-2\mathit{x}+4>0\to {\mathit{x}}^2-2\mathit{x}+1+3>0$
${\left(\mathit{x}-1\right)}^2+3>0$
${\left(\mathit{x}-1\right)}^2>-3\to$ Always True because positive real number is always greater than negative real number.