Given $b$ and $c$ with $b,c>1$, is it possible to construct a polynomial $p(x)$, whose degree $n$ depends on neither $c$ nor $b$, such that:

• $p$ is either non-negative and strictly increasing or negative and strictly decreasing on $[1,c]$

• and $|b \cdot p(c)| < |p(0)|$?

This might be satisfied by an interpolating polynomial, but how to actually construct it is beyond me.

Given $b$ and $c$ with $b,c>1$, is it possible to construct a polynomial $p(x)$, whose degree is $n$ for all $c$ and $b$, such that:

• $|p|$ is strictly increasing on $[1,c]$

• and $|b \cdot p(c)| < |p(0)|$?

This might be satisfied by an interpolating polynomial, but how to actually construct it is beyond me.

I was playing around with polynomials a few days ago, mostly about bounds, and I had this question: Given $b$ and $c$ such that $b,c>1$ and $b,c \in \mathbb{R}$, is it possible to construct a polynomial $p(x)$, whose degree $n$ depends on neither $c$ nor $b$, that is either non-negative and strictly increasing or negative and strictly decreasing (but not both) on $[1,c]$, and that $|b \cdot p(c)| < |p(0)|$? I honestly don't know where to begin-- the best I've gotten is that it would probably be an interpolating polynomial (how to actually construct it is beyond me).

Given $b$ and $c$ with $b,c>1$, is it possible to construct a polynomial $p(x)$, whose degree $n$ depends on neither $c$ nor $b$, such that:

• $p$ is either non-negative and strictly increasing or negative and strictly decreasing on $[1,c]$

• and $|b \cdot p(c)| < |p(0)|$?

This might be satisfied by an interpolating polynomial, but how to actually construct it is beyond me.