For all a, b ∈ R, the function $\mathrm{f}\left(\mathrm{x}\right)=3{\mathrm{x}}^4-4{\mathrm{x}}^3+6{\mathrm{x}}^2+\mathrm{ax}+\mathrm{b}\mathrm{\hspace{0.17em}}$ has

$\begin{array}{l}{\mathrm{f}}^1\left(\mathrm{x}\right)=12({\mathrm{x}}^3-{\mathrm{x}}^2+\mathrm{x})-\mathrm{a}\\ {\mathrm{f}}^{11}\left(\mathrm{x}\right)=12(3{\mathrm{x}}^2-2\mathrm{x}+1)>0\\ \Rightarrow {\mathrm{f}}^1\left(\mathrm{x}\right)\mathrm{is}\mathrm{an}\mathrm{icreasing}\mathrm{function}\end{array}$
$\therefore {\mathrm{f}}^1\left(\mathrm{x}\right)\text{ is polynomial of degreethere. }{\mathrm{f}}^1\left(\mathrm{x}\right)=0$
has exactlyone real root, so function has exactly one extremum for all real values of a & b.