Monotonic 3rd order polynomial function condition

The monotonic function is the function that has its derivative:

$f'(x)<0$ for all real $x$ values

or

$f'(x)>0$ for all real $x$ values

In other words, $f'(x)=0$ must not have real roots

$f'(x)=3x^2+2ax+3$

For ( $3x^2+2ax+3=0$ ) not to have any real roots, this condition must be met:

$4a^2-36<0$

$a^2-9<0$

$|a|<3$

So the solution is:

$k=3$