Show that every real polynomial $x\in C[a,b]$ of degree $n\ge 1$ with leading term $\beta_n t^n$ satisfies $$||x||\ge |\beta_n|\frac{(b-a)^n}{2^{2n-1}}.$$
I am having difficulty proving this. Here on wikipedia it is shown how to do this when $a=-1$ and $b=1$, but how can it be shown for arbitrary $a$ and $b$?