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Understanding when a function is increasing or decreasing is crucial to graphing and analyzing the behavior of the function. A function is said to be increasing on an interval if the output values increase as the input values increase; conversely, it is decreasing if the output values decrease as the input values increase.

Based on the function provided, f(x)=x^{3}-3 x^{2}+2 , we use its derivative f'(x)=3x^{2}-6x to determine its behavior. This derivative allows us to check the growth rate of f(x) at any value of x . When f'(x) > 0 , the function is increasing, while f'(x) < 0 indicates a decreasing interval.

Through testing intervals around the critical points, which are the x-values that make the derivative zero or undefined, we can see that the function is increasing on the intervals (-∞,0±Õ and (2,+∞) , and decreasing on (0,2] . These intervals are pivotal for graphing functions and understanding their overall behavior.

The concept of critical points is essential in calculus for understanding the local behavior of functions. Critical points occur where the derivative of a function is zero or undefined. They are potential locations of relative maxima, minima, or points of inflection.

For the given function f(x)=x^{3}-3 x^{2}+2 , the critical points were found by setting the derivative f'(x)=3x^{2}-6x equal to zero and solving for x . This resulted in two critical points x=0 and x=2 .

Locating these critical points enables us to assess where the function may change its rate of increase or decrease. It's crucial to understand not every critical point will correspond to a peak or a trough on the graph; some may be points of inflection where the concavity changes. Thematically, identifying critical points is akin to a detective interrogating suspects to deduce the storyline of the function's graph.

The First Derivative Test is a profound tool in calculus for determining whether a function has a relative maximum, minimum, or neither at a critical point. Once critical points are identified, this test uses the sign of the derivative before and after these points to conclude the function's behavior.

In practice, choose a value to the left and to the right of a critical point and evaluate the derivative at these points. An intuitive way to remember is if the derivative changes from positive to negative as you pass through the critical point, the function has a peak (relative maximum). Conversely, if the derivative changes from negative to positive, the function has a trough (relative minimum).

For function f(x) , applying the First Derivative Test around the critical points x=0 and x=2 , we can predict the function's behavior without seeing the graph. At x=0, the test indicates a change from increasing to decreasing, suggesting a relative maximum. At x=2, the increasing behavior resumes, implying a relative minimum. These insights form a baseline for studying calculus and for appreciating the beauty of functions' behaviors.