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The Second-Derivative Test is a useful tool for determining the behavior of a function at its critical numbers. Once you've found a critical number where the first derivative is zero or undefined, the second derivative can help discern whether this point is a relative maximum or minimum. Here's how it works:

• If the second derivative at a critical number is greater than zero, the function is concave up at that point, indicating a relative minimum.
• If the second derivative is less than zero, the function is concave down, suggesting a relative maximum.
• If the second derivative is zero, the test is inconclusive. In such cases, other methods such as the first-derivative test might be needed to draw a conclusion.

Applying this to our exercise, we found the second derivative of the given function to be \( f''(x) = 12x^2 - 4 \). This second derivative was used to test each critical number and determine the type of extremum present at those points.

A relative maximum refers to a point on a function where its value is higher than all nearby points, effectively forming a peak.

In the example problem, we identified one such relative maximum at the critical number \( x = 0 \). The second derivative at this point, \( f''(0) = -4 \), is less than zero, indicating concavity downwards. This downward bowl shape confirms that the function peaks at this critical point, resulting in a local maximum.

Understanding where relative maxima occur provides insight into the behavioral peaks of the function, which is essential for graphing and optimization problems.

Conversely, a relative minimum is characterized by a point where the function dips lower than all nearby points, forming a valley.

In the given problem, the critical numbers \( x = 1 \) and \( x = -1 \) were identified as having relative minima. For these points, the second derivative, \( f''(1) = 8 \) and \( f''(-1) = 8 \), were both positive. This positive value tells us that the function is concave up at these points, forming a cup-like shape indicative of a local minimum.

Recognizing and understanding relative minima is crucial for tasks that involve finding the lowest possible values in a set interval, important for applications such as cost minimization or efficiency maximization.