I am trying to solve this problem, but I don't have any idea. Maybe it doesn't look at first sight that Lagrange interpolation can be used, but I found this problem in that chapter of Numerical methods, so I suppose that interpolation can be used.
The problem is:
Let $x_1 > x_2 > ... > x_n$ be natural numbers and let $P(x) = x^n + a_1 x^{n-1}+...+a_n$ be an arbitary polynomial with real coefficients. Prove that there exists $i\in {{1, 2, ..., n}}$ so that $|P(x_i)|\geq \frac{n!}{2^n}$.
Does anyone have idea? Thanks in advance.