I wish to calculate the maximum value of the product $(x-x_0)(x-x_1)\cdots(x-x_n)$ where $x$ lies in the interval $[x_0,x_n]$. I know to apply the AM-GM inequality to get $$(x-x_0)(x-x_1)\cdots(x-x_n) \le \left(\frac{(x-x_0)+(x-x_1)+\cdots+(x-x_n)}{n+1}\right)^{n+1}$$
However, how do I find at what $x\;\epsilon\;[x_0,x_n]$ this RHS becomes maximum? Since this is mostly a long product, I do not have the option of differentiating and finding the roots. Or is there any different method?
Edit: The absolute value of the product is to be considered. So I guess, I need to find the maximum or minimum value of the product in the given interval. I did not notice that I needed to consider absolute values, sorry about that. I need the maximum value of $$|(x-x_0)(x-x_1)\cdots(x-x_n)|\;\;\;\;x\;\epsilon\;[x_0,x_n]$$
If it helps, I noticed that for $|(x-x_0)(x-x_1)|$, the maximum value in the given interval occurs at $x=\cfrac{x_0+x_1}{2}$ and hence, the maximum value of the product is given as $$|(x-x_0)(x-x_1)|\le\cfrac{(x_1-x_0)^2}{4}$$ Can this be extended to more terms in the product in any way?
If you apply AM-GM inequality on numbers, say, $a_1,a_2,a_3 \dots a_n$; then AM$=$GM iff $$a_1=a_2=a_3=\dots =a_n$$
But unfortunately, $x-x_i=x-x_j$, results only if $x_i=x_j ~ ;~\forall i,j \in \{0,1,2 \dots n\}$, which is not (probably) the case here. Hence, derivatives is the only possible way to determine the maximum value of product.