How to find number of solutions using the derivative?

Question

I know these are probably well-known results, but I want to find how many roots a function has using its derrivatives.
Consider a two times differentiable function $f:\mathbb{R} \to \mathbb{R} $. If $f''(x) =0$ has only one root, does this tell me anything about how many roots $f(x) =0$ has? Can this be generalised?

Answer 1

For simplicity, let's imagine that the roots of $f$ are all distinct.

Suppose the function had $4$ distinct roots, $a<b<c<d$. Then we know that $f'(x)$ has a root in $(a,b)$, another in $(b,c)$, and a third in $(c,d)$. Between any two roots of $f'$ there is a root of $f''$. Thus, in this situation there would be at least $2$ roots of $f''$. Thus $f(x)$ can have at most $3$ roots. The cubic $f(x)=x^3-x$ has three distinct roots and $f''(x)=6x$ has only a single root, so this bound is sharp.