Let $f:[1008, 1009] \to \Bbb R$ be defined by $$f(x) =|x| \cdot |x-1| \cdot |x-2| \cdots |x-2017|$$ Find the maximum of $f(x)$ without using derivatives.
2026-02-22 19:33:10.1771788790
Maximum of a product of absolute values
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By AM-GM $$f(x)=x(x-1)...(x-1008)(1009-x)...(2017-x)=$$ $$=x(2017-x)\cdot(x-1)(2016-x)\cdot...\cdot(x-1008)(1009-x)\leq$$ $$\leq\left(\frac{2017}{2}\right)^2\cdot\left(\frac{2015}{2}\right)^2\cdot...\cdot\left(\frac{1}{2}\right)^2=\frac{(2017!!)^2}{2^{2018}}$$ The equality occurs for $x=2017-x$ or $x=1008.5$