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To identify critical points and analyze how a function behaves at those points, the first derivative test is an essential tool. It focuses on the change of direction in the slope of the function's graph. When the first derivative of the function, represented as \(f'(x)\), transitions from positive to negative, the function is said to have a relative maximum at that point. Conversely, if \(f'(x)\) changes from negative to positive, the point is a relative minimum.

In the exercise, after factoring the first derivative to find \(x = 0\) and \(x = \frac{1}{2}\), we observed the sign change of \(f'(x)\) around those points. For \(x = 0\), since \(f'(x)\) switched from positive to negative, it indicated a relative maximum. This change in the sign of the slope, detailed in step 4, is what confirms the behavior at that critical point without needing the second derivative test.

It's critical to examine the intervals around the critical points when applying this test. If the derivative does not change sign, then the critical point does not represent a relative extremum.

While the first derivative test uses the sign of the derivative to infer the nature of critical points, the second derivative test uses the curvature of the function, as indicated by the sign of the function's second derivative, \(f''(x)\). This test can quickly determine whether a critical point is a concave up (relative minimum) or concave down (relative maximum).

For the function in our exercise, the second derivative was \(f''(x) = 36x^2 - 12x\), in which we substituted the critical points to evaluate it. At \(x = \frac{1}{2}\), the second derivative is positive, indicating a relative minimum because the function is concave up. However, at \(x = 0\), \(f''(0) = 0\), rendering the second derivative test inconclusive; a different approach—like the first derivative test—is required in such cases.

The second derivative test is favored for its simplicity, but it’s important to be aware of its limitations and be prepared to use the first derivative test when needed.

Relative extrema refer to the high and low points on a function within a specific interval. These points are categorized as relative maxima or minima based on their heights compared to nearby points on the function. Identifying relative extrema is crucial in understanding the overall shape and behavior of functions.

In the context of the exercise, the function displayed two critical points, one of which was a relative maximum at \(x = 0\), and the other a relative minimum at \(x = \frac{1}{2}\). These findings derive from comprehensively applying the first and second derivative tests. Recognizing these extrema proves valuable in scenarios like optimizing real-life systems or examining the turning points on a graph.

When determining relative extrema, it's imperative to examine the function comprehensively in the neighborhood of critical points to correctly infer the behavior and ensure the precise classification of each extremum.