`f(x) = (x - 3)^(1/3)` Find the critical numbers, open intervals on which the function is increasing or decreasing, apply first derivative test to identify all relative extrema.

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to evaluate the critical values, hence, in order to do so, you must solve for x the equation `f'(x) = 0` .

You need to find `f'(x), ` such that:

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

You need to evaluate `f'(x) = 0` :

`1/(3*(root(3)(x-3)^2)) = 0`

There is no value of x to verify the equation  `f'(x) = 0` , hence, there are no critical points.

Notice that the derivative is positive on R, hence, the function is increasing on R.