$f$ is a polynomial of degree $n\ge1$ and $\forall x,x\in \Bbb R \rightarrow f(x)\in\Bbb R$.
Prove that: (a)$f$ has at most one more real root than $f'$ (b)$f'$ has no more non-real roots than $f$ (c)If all roots of $f$ are real, then all roots of $f'$ are real.
I notice that (a) and (b) are equivalent, so if I can prove (a), (b) is automatically proven (because $f'$ has exactly n-1 roots). But I am just totally unaware of the way to start.
Based on the homework context, I think it has something to do with the following: Theorem: Let $z_0$ be a root of multiplicity m of polynomial f: (1)If m=1, then f'($z_0)\neq0$. (2)If m>1, then $z_0$ is a root of f', of multiplicity m-1.