Question

f(x)=x^2 -1 / x^3
on what intervals is the function increasing and desreasing?

Answer 1

In the previous question you posted you found the first derivative and the critical points.
Based on the sign of f'(x) at various intervals decided by the critical point and the point of discontinuity of the function you can find if the function is increasing or decreasing.

Answer 2

The critical points were -sqrt(3) and sqrt(3).
The function is not defined at x = 0
Split the domain into the following intervals:
-inf -sqrt(3) 0 sqrt(3) inf
choose a suitable point in each interval and find f'(x).
If f'(x) is positive, the function is increasing in that interval
If f'(x) is negative, the function is decreasing in that interval