Problem: Let $P(x)$ be a monic polynomial of degree $n$ with real coefficients. Prove that it is not possible that for all $t \in [-1, 1]$, $$\frac{-1}{2^n} < P(t) < \frac{1}{2^n}.$$
I tried it to put it in another way: If $P(a)$ and $P(b)$ are the maximum and minimum respectively of $P$ on $[-1, 1]$, then show that $$P(a)-P(b) > \frac{1}{2^{n-1}}.$$
But I am completely stuck.