Final answer:
To find intervals where the function f is increasing, determine its derivative, set it greater than zero, find critical points, test intervals with a point, and list those where the derivative is positive. It's essential to follow steps accurately and consider the function's domain.
Step-by-step explanation:
To determine the intervals where the function f is increasing, we must analyze the function's derivative. An increasing function is one where, as x increases, f(x) also increases. The derivative f'(x) represents the rate of change of the function; if f'(x) is positive over an interval, then f is increasing on that interval.
We begin by finding the derivative of the function f with respect to x. Once the derivative is obtained, we set it greater than zero to find where the function is increasing. Then, we solve for x to find the critical points and use these points to test the intervals.
To check each interval, we pick a test point from within the interval and substitute it into the derivative f'(x). If the result is positive, the function is increasing on that interval. Lastly, we list all the intervals where the derivative is positive, indicating where the function is increasing.
Keep in mind that each step is important for accuracy. Skipping steps or misinterpreting the derivative can lead to incorrect intervals. Moreover, it's crucial to consider the domain of the function, as some functions may not exist or may not have a derivative at all points.