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In calculus, critical numbers play a pivotal role in understanding the behavior of functions. They refer to the points on the function where the first derivative is either zero or undefined. Identifying these numbers is crucial for analyzing a function's increasing or decreasing trends and can indicate potential local maximums and minimums—a key aspect of the First Derivative Test.

To find critical numbers, one must calculate the derivative of the function and set this value to zero. In cases of piecewise functions, like the provided exercise, each segment of the function must be treated individually to find derivatives and, consequently, the critical numbers within their respective intervals. In the exercise, the critical numbers 0 and 1 were identified by setting the derivatives of the two pieces to zero. These points, along with any points where the derivative does not exist, partition the entire number line into intervals where the function’s behavior can be analyzed further.

It is worth noting that not all critical numbers will result in a relative maximum or minimum. To determine this, further tests, such as the First Derivative Test, are required. This will indicate whether each critical number correlates to a peak, trough, or neither in the function's graph.

Piecewise functions are defined by multiple sub-functions, each applicable to a specific interval within the domain. They are particularly useful for modeling situations where a function's rule changes based on the input value. In calculus, they pose a unique challenge when determining derivatives, as each piece must be handled separately.

For instance, in our exercise, the function is represented differently for values less than or equal to zero and for values greater than zero. Thus, to find the derivative, we must derive each piece of the function individually, leading to two separate derivative functions that are defined on their respective intervals. When working with piecewise functions, attention must be given not just to the process of derivation but also to the continuity and differentiability at the points where the pieces of the function meet. This can often lead to discovery of critical numbers and is vital for understanding the overall behavior of the function across its entire domain.

The First Derivative Test is a fundamental tool used to determine where a function is increasing or decreasing. After the critical numbers of a function are found, one can use these points to test the intervals between them to see if the function’s slope is positive (increasing) or negative (decreasing).

In our exercise, once the critical numbers 0 and 1 were determined, the real number line was divided into three intervals: (-∞, 0], (0, 1), and (1, ∞). By choosing test points within these intervals and substituting them into the derivative, we can determine the function's behavior on each interval. For example, in the given interval (-∞, 0] the derivative's value was positive, indicating that the function is increasing on this interval. Similarly, for (0, 1), the function continues to increase. However, for the interval (1, ∞), the derivative has negative values, indicating the function is decreasing therein.

The importance of understanding increasing and decreasing intervals lies in predicting the function’s graph and potential extremum points where local minimums and maximums occur. It opens the path to comprehensively assessing a function’s characteristics, which is essential in calculus and applications involving optimization problems.