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[SOLVED] x-coordinates of Optimization Points for Polynomial Function: f(x)=(x-1)^4+(x-2)^4+...+(x-n)^4

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,588
For an integer $n\ge 2$, find all real numbers $x$ for which the polynomial $f(x)=(x-1)^4+(x-2)^4+\cdots+(x-n)^4$ takes its minimum value.
 

castor28

Well-known member
MHB Math Scholar
Oct 18, 2017
242
For an integer $n\ge 2$, find all real numbers $x$ for which the polynomial $f(x)=(x-1)^4+(x-2)^4+\cdots+(x-n)^4$ takes its minimum value.

Since $x=\dfrac{n+1}{2}$ is an axis of symmetry, the point $x=\dfrac{n+1}{2}$ is either a minimum of a maximum, depending on the shape of the quartic.

However, the derivative $f'(x) = 4\left((x-1)^3+\cdots+(x-n)^3\right)$ is an increasing function (since it is a sum of increasing functions). Therefore, $f'(x)$ as exactly one zero, and $f(x)$ has only one critical point.

We conclude that $x=\dfrac{n+1}{2}$ is the unique global minimum.