find the partition number

Question

f(x)= x^1/3

-> f'(x)= 1/3x^-2/3

How can I find the partition number and critical number?

Answer 1

Def 1: Critical numbers are numbers in the domain of the function and when the first derivative equals 0 or undefined.

Def 2: The partition numbers are numbers when the first derivative equals 0 or undefined.

Solution:

There are no real numbers where derivative f'(x)= 1/3x^-2/3 equals 0, but there is one where derivative does not exist, it is x=0.

Thus x=0 is a partition number.

Also, x=0 belongs to the domain of the function f(x)= x^1/3, so it is a critical number too.

Answer: x=0 is a critical and a partial point of f(x)= x^1/3.