For all  $a,b\in R$   the function   $f\left(x\right)=3x^4-4x^3+6x^2+ax+b$  has

Given $f\left(x\right)=3x^4-4x^3+6x^2+ax+b$

f′(x)=$12x^3-12x^2+12x+a$
And 
f′′(x)=$36x^2-24x+12$
=12($3x^2-2x+1$)
Now 
D=$b^2-4ac=2^2-4\left(3\right)$
=4−12
=−8
Hence
D<0, thus 
f"(x)>0 for all x.
Hence f(x) will only have minima, and no maxima.
Also f"(x) has no real roots.
Therefore f′(x), x will have atmost one real root.