Rolle's Theorem states that for any function continuous on an interval $[a,b]$, and differentiable on its interior, if $f(a) = f(b) = 0$, then $\exists c ∈ [a,b]$ such that $f'(c) = 0$.
I believe that we can therefore conclude that,
(1) If $f(x)$ has $n$ roots, then $f'(x)$ has at least $n-1$ roots,
(2) If $f'(x)$ has $n-1$ roots, then $f(x)$ can have at most $ n$ roots
Is that true?
Also, are these two statements enough to encompass all possibilities, given information about the roots of either f(x) or f'(x)?
For instance, if we know $f(x)$ has at least $3$ roots, then we can conclude that $f'(x)$ must have at least $2$ roots from $(1)$. Or, if we are told that $f'(x)$ has at most $4$ roots, $f(x)$ cannot have more than $5$ from (2).
Are they any possibilities that would be left out from only these $2$ cases?
Thanks!
Yes, what you say is true and it is very useful to have that in mind, well done ! But be careful, $f$ also has to be differentiable on $]a, b[$.
A good application of this is for real polynomials. For instance, it is a good exercice to show that if $P\in\mathbb{R}[X]$ has $\deg P$ real roots counted with multiplicity, then $P'$ has $\deg P - 1$ real roots counted with multiplicity.