A way to find the total variation of a polynomial if the zeroes of the derivative are known.

Question

I came across a question in Apostol's book which said to describe a method to find the total variation of a polynomial if the zeroes of the derivative of it is known ( points at which the derivative is zero). Intuitively, I suppose the highest variation can be obtained by taking a partition that contains all the points at which the derivative is zero. But I find it hard to prove this. Any hints will be welcomed. Thanks

Answer 1

Let $P(x)$ be a polynomial, and let $x_1<x_2\dots< x_N$ be the zeroes of $P'$ on a given interval $[a,b]$, that for simplicity, I will assume are simple. Then $x_1,\dots x_N$ are local maxima or minima, and $P$ is strictly monotone on each interval $[x_i,x_{i+1}]$, $1\le i<N$, on $[a,x_1]$ and on $[x_N,b]$. The total variation of $P$ on $[a,b]$ is then $$ |P(a)-P(x_1)|+\sum_{i=1}^{N-1}|P(x_i)-P(x_{i+1})|+|P(x_N)-P(b)|. $$

I leave you the case in which the zeroes have multiplicities larger than $1$.