Let $x_i=\frac{i}{n}\;, i=0,1,...,n\;$ be $\;n+1$ equidistant points on the interval $[0,1]$. Prove: The maximum value of: $$\prod_{i=0}^n|x-\frac{i}{n}|$$ in the interval $[0,1]$ is achieved for $x\in (0,\frac{1}{n})$ or $x\in (\frac{n-1}{n},1)$ (i.e in the first or last subinterval).
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3This is false for $n=3$. The values in the first and last interval are negative, whereas the values in the middle interval are positive, so the maximum is in the middle interval. – joriki Dec 27 '19 at 12:32
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@joriki Sorry, I meant the absolute value. I will edit my question. – user401516 Dec 27 '19 at 12:34