analytical form for extreme points of polynomial on intervals.

Question

Given coefficient $w_0, w_1, \cdots w_d$ of d degree polynomial $f(x) = \sum_{i=0}^d w_ix^i$.

And some interval $x \in [x_1, x_2] \subset \mathbb{R}$, I'm interested in the extreme points of this convex set: $CH(\{(x, f(x))| x \in [x_1, x_2]\})$. And here $CH$ is the convex hull operation.

I'm wondering if there is any analytical form to enumerate all the extreme points.

Would $\{x|f'(x)=0 \} \cap [x_1, x_2] \cup \{ x_1, x_2\}$ this be the set of $x$ axis for extreme points?

Answer 1

As mentioned in the comments, this isn't meant to be an answer, this is more like an extended comment.

I'll take as an example $f(x) = x^{3}-x^{2}-x+1$ considered in the interval $[-1,1]$.

The convex hull of the curve is orange-shaded. The definition of extreme points of the convex hull is simply that they are points that are not on any line segment between other points of the convex hull. For our example, we get

The set of extreme points is $\{ (x,f(x)) | -1 \leqslant x \leqslant 0 \} \cup (1,0)$

Generally speaking, we have

• $(x_1,f(x_1))$ and $(x_2,f(x_2))$ are always members of the set of extreme points (in this context).
• if the second derivative doesn't change signs over the $[x_1,x_2]$ interval, then $f$ is either convex or concave and the set of extreme points is the entire curve over that interval. As a corollary, for polynomials of degree $2$, this is the case for all $x_1,x_2$.

Things get "hairy" when the segment contains both convex and concave portions. Following example is for $g(x) = x^{4}+x^{3}-x^{2}-x+1$ considered around $[-3/2, 1]$

The point I want to make here, is that the common tangent does not touch the curve at stationary points (grey points) of the curve (as you seem to have hoped). And from a general point of view, it seems hard to avoid ODEs on a case-by-case basis in order to make an exact determination for the set of extreme points of the convex hull of the curve.

I'd be delighted if I'm wrong, but for now, I don't see a path to a general analytical form.