You've correctly used the quotient rule to find the derivative or gradient of $f(x)$.
The question then asks for two things: critical points and open intervals on which the function is either increasing or decreasing.
For any point $x$ in the domain of the function, one of three things is true. Either:
- $f'(x) > 0$, i.e. the function is increasing;
- $f'(x) = 0$, i.e. the function is a critical point;
- $f'(x) < 0$, i.e. the function is decreasing.
You can find the points which fall into category 2; any other points will fall into open intervals, each of which will either satisfy category 1, increasing, or category 3, decreasing.
If you take your domain, the reals, and remove the critical points, you'll be left with just open intervals. Each of these has $f(x)$ either increasing on all of it or decreasing on all of it. This is because your derivative is a continuous function, meaning that it can't jump from being negative (category 3) to being positive (category 1) without being 0 at some point in between.