# How do you find the critical numbers for f(x)= x^(-8) ln x to determine the maximum and minimum?

A critical number for $f$ is a number $c$ in he domain of $f$ where $f ' \left(c\right) = 0$ or $f ' \left(c\right)$ does not exist.
$f \left(x\right) = {x}^{-} 8 = \frac{1}{x} ^ 8$ has domain: all reals except $0$.
$f ' \left(x\right) = - 8 {x}^{-} 7$ is never $0$ and fails to exist only at $0$, which is not in the domain of $f$. So, $f$ has no critical numbers.