My task is to prove: $$ |\prod_{i=0}^n(x-x_i)|\leq\frac{n!}{4}(\frac{b-a}{n})^{n+1} $$ where $x_i=a+i\frac{b-a}{n}$ for $i=0,...,n$ and $x\in [a;b]$
I managed to prove that: $$ |(x-x_0)(x-x_1)|\leq \frac {1}{4}(x_1-x_0)^2 $$ so it's for $n=1$, using a.m.-g.m., but I don't know in which direction to go in order to prove it for all $n$