Question

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

Answer 1

The function has no critical numbers.

Explanation:

A critical number for `f` is a number `c` in he domain of `f` where `f'(c) = 0` or `f'(c)` does not exist.

`f(x) = x^-8 = 1/x^8` has domain: all reals except `0`.

`f'(x) = -8x^-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.