1.5 .8 ** Let $m, n$ be two distinct positive integers, and real number $x \in(0,1)$. Find the maximum value of the expression $\left|x^{m}-x^{n}\right|$.
Assume $m>n$, let $f(x)=\left|x^{m}-x^{n}\right|=x^{n}-x^{m}$, then
$$
f^{\prime}(x)=n x^{n-1}-m x^{m-1}
$$

is negative at $x=x_{0}$.
Therefore, when $x=\left(\frac{n}{m}\right)^{\frac{1}{m-n}}$, $f(x)$ reaches its maximum value. Similarly, for the case $m<n$, we can obtain that the maximum value of $\left|x^{m}-x^{n}\right|$ is $|m-n|\left(\frac{n^{n}}{m^{m}}\right)^{\frac{1}{m-n}}$.