>Solve the following equation on the set of real numbers R ${{\left| x-2017 \right|}^{2017}}+{{\left| x-2018 \right|}^{2018}}=1$.

Question

Solve the following equation on the set of real numbers R: $${{\left| x-2017 \right|}^{2017}}+{{\left| x-2018 \right|}^{2018}}=1$$

Answer 1

If $x<2017$, $|x-2018|>1$ and hence $|x-2017|^{2017}+|x-2018|^{2018}>1$.

If $x>2018$, $|x-2017|>1$ and hence $|x-2017|^{2017}+|x-2018|^{2018}>1$.

If $2017<x<2018$, $|x-2017|,|x-2018|\in(0,1)$ and hence

$$|x-2017|^{2017}+|x-2018|^{2018}<|x-2017|+|x-2018|=x-2017+2018-x=1$$

$x=2017$ and $x=2018$ are the only solutions.