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Understanding the monotonicity of a function—a quality describing how a function moves strictly in one direction (either always increasing or always decreasing) within certain intervals—is largely based on calculating derivatives. The derivative of a function at a point gives the slope of the tangent line to the function at that point. When we want to determine if a function is increasing or decreasing on an interval, we look at the sign of its first derivative, denoted as \(f'(x)\).

For instance, if \(f'(x)>0\) on an interval, the function is ascending, whereas \(f'(x)<0\) denotes a descending function. By examining the derivative, one can comprehend the behavior of the function in terms of monotonicity, which ascertains whether the function is exhibiting consistent incremental or decremental movement.

Critical points are pivotal in understanding the features of functions. They are the x-values where the first derivative of the function either equals zero or is undefined. These points potentially mark where the function could have a relative maximum or minimum or where a change in the direction of the graph occurs—a shift from increasing to decreasing or vice versa.

To find such points, we set the first derivative equal to zero and solve for \(x\). Occasionally, these solutions might require further analysis, especially if they yield multiple critical points or involve higher-degree polynomials, where factoring or using the quadratic formula might be necessary. Identifying these points accurately is essential as they lay the groundwork for the next steps in characterizing the function's behavior.

Once critical points are calculated, it's key to determine their impact on the function's behavior. Interval testing is a method where you pick test points from each interval created by the critical points to evaluate the function’s slopes—via its derivative—within those intervals.

To conduct interval testing, select any value within an interval and plug it into the first derivative. The sign of \(f'(x)\) at the test point will reflect the behavior of the function throughout that entire interval. If \(f'(x)\) is positive at the test point, the function is increasing on that whole interval, and if \(f'(x)\) is negative, the function is decreasing. This process must be done for each interval to map the complete behavior of the function accurately over its domain.