The steps for determining intervals are given as under :

1: Determine derivative of given function i.e. f’(x).

2: Determine sign of derivative in different intervals.

3: Determine monotonic nature of function in accordance with following categorization :

5: The interval is open “( )” at end points, if function is not continuous at end points. However, interval is close “[]” at end points, if function is continuous at end points.

In order to illustrate the steps, we consider a function,

Its first derivative is :

Here, critical point is 1/2. First derivative, f’(x), is positive for x>1/2 and negative for x<1/2. The signs of derivative are strict inequalities. It means that function is either strictly increasing or strictly decreasing in the open intervals. We know that infinity end is an open end. But, function is continuous in the given interval. Hence, we can include end point x=1/2. Further, since derivative is zero at x=1/2 i.e. at a single point, function remains strictly increasing or decreasing.