For a given real polynomial $$p(x)=a_0 + a_1x + a_2x^2+\dots +a_nx^n,\quad\text{with $a_0=0$, $x\in \mathbb{R}$},$$ is there any proof that all local extrema will always lie between two roots of $p(x)=0$?
I was messing around with some visualizations and tried to proof it using derivatives but could not solve it. Internet search wasn't helpful either. Thanks for your help!
That is not true. Take for example $$p(x)=6\int_0^x (t-1)(t-2)dt=2x^3-9x^2+12x$$ It has local extrema at $1$, and at $2$ but $p$ has just one root, that is $0$.
However, by Gauss-Lucas theorem, for any non-constant polynomial $p$, all zeros of a $p'$ (the derivative is zero at local extrema) belong to the convex hull in the complex plane of the set of complex zeros of $p$.