TITLE: What is so stunning or elegant about this theorem?
QUESTION [7 upvotes]: I read an essay by a mathematician (whose name I forgot) who visited TIFR in India and he was writing about his mathematical experiences during the visit. In his essay the mathematician had written that he was stunned by the elegance and beauty of the following theorem.
Let $a_i$ be integers and
$$
f(x) = x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \cdots + a_2 x^2 + a_1 x + a_0,
$$
then $$\displaystyle\max_{x\in(-2,2)} |f(x)| \ge 2.$$
Questions:

What is so stunning or elegant about it?
I am looking for a proof
Any reference to this theorem in literature?

Note: I am trying to recall and search the essay. I shall update my post with its link if I am able to find it.

REPLY [4 votes]: By Chebyshev's Theorem (see Chebyshev's Theorem regarding real polynomials: Why do only the Chebyshev polynomials achieve equality in this inequality?), if $p(x)$ is a real polynomial of degree $n\geq 1$ with leading coefficient $1$ then
$$\max_{x\in [-1,1]}|p(x)|\geq \frac{1}{2^{n-1}}.$$
Now $p(x):=f(2x)/2^n$ is a polynomial with leading coefficient $1$ and it follows that
$$\max_{x\in [-2,2]}|f(x)|=\max_{x\in [-1,1]}|f(2x)|=2^n\cdot\max_{x\in [-1,1]}|p(x)|\geq \frac{2^n}{2^{n-1}}=2.$$
P.S. As far as I can see, here we do not need that the coefficients $a_i$ have to be integers.