Are there any increasing or decreasing intervals of a quadratic equation with no real values of $x$?

Question

Using first derivative test, find the increasing/decreasing intervals and local maximum and local minimum values of the function $f(x) = x^3 -3x^2 - 6x$

$$f'(x)= 3x^2 -6x -6=0 $$ what do I write for this type of question ?

Answer 1

$$f(x)=x^3-3x^2-6x$$

$$f'(x)=3x^2-6x-6=0$$

$$f'(x)=x^2-2x-2=0$$

$$D=(-2)^2-4\times 1\times (-2)=12$$

$x_1=1-\sqrt{3}$, $x_2=1+\sqrt{3}$

The function is increasing (because $f'(x)>0$) in $(-\infty,1-\sqrt{3})$ and increasing (because $f'(x)>0$) in $(1+\sqrt{3},\infty)$ , decreasing (because $f'(x)<0$) in $(1-\sqrt{3},1+\sqrt{3})$

And there is a local minimum for $x=1+\sqrt{3}$, a local maximum for $x=1-\sqrt{3}$.