Using Rolle's theorem prove that if $b^2<3ac$, then there is exactly one root to $f(x) \equiv ax^3+bx^2+cx+d=0$
I literally have no idea how to use Rolle's Theorem.
Rolle's Theorem: if the function $f \in C[a,b]$ and differentiable on $(a,b)$, and $f(a)=f(b)$, then there exists $c \in (a,b)$ for which $f'(c)=0$.
What I did so far:
$D=4b^2-12ac>4b^2-4b^2=0 \implies f'(x)=0$ has exactly two roots.
If I use given inequality, I don't prove anything what I need. Any help would be appreciated.
I have looked at a similar example Prove that if ab > 0 then the equation $ax^3 + bx + c = 0$ has exactly one root by Rolle's theorem but I didn't find it a lot helpful.