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Relative extrema are points on the graph where a function reaches a local maximum or minimum value. These points are important for understanding the behavior of the function. To find relative extrema, one common method is the first-derivative test.

When the first derivative of a function changes sign, it indicates a relative extremum. For example, if the derivative changes from positive to negative, the function has a relative maximum at that point. Conversely, if it changes from negative to positive, there is a relative minimum. In the given problem, we use this concept to find where the graph of the function has these turning points.

These points are valuable in many real-world applications like finding the highest points (peaks) on a road or the lowest points (valleys) in business profit graphs. Understanding relative extrema helps in making informed decisions based on these peaks and valleys.

The power rule is a fundamental technique in calculus for finding the derivative of a function of the form \(f(x) = x^n\), where \(n\) is any real number. The power rule states:

\(f'(x) = nx^{(n-1)}\)

Applying the power rule makes it easy to compute derivatives quickly. For example, in the exercise, the function is given as \(f(x) = x^3 - 6x^2 + 1\). Using the power rule, we find:

• For \(x^3\), the derivative is \(3x^2\)
• For \(-6x^2\), the derivative is \(-12x\)
• For the constant term \(1\), the derivative is \(0\)

Therefore, the first derivative is \(f'(x) = 3x^2 - 12x\). By simplifying the process of finding derivatives, the power rule lets us focus on higher-level problem-solving strategies.

A variation chart is a valuable tool for understanding how the sign of the first derivative changes over different intervals. This helps to locate relative extrema. We set up the variation chart by:

• Identifying the critical points where \(f'(x) = 0\)
• Dividing the number line into intervals based on these critical points
• Testing a point within each interval to determine whether \(f'(x)\) is positive or negative

In our problem, the critical points are \(x=0\) and \(x=4\). We test points in the intervals \((-∞, 0)\), \((0, 4)\), and \((4, ∞)\):

• For \(x = -1\), \(f'(-1) > 0\)
• For \(x = 2\), \(f'(2) < 0\)
• For \(x = 5\), \(f'(5) > 0\)

This chart shows us that \(f'(x)\) changes from positive to negative at \(x=0\) (a relative maximum) and from negative to positive at \(x=4\) (a relative minimum). Visualization with a variation chart simplifies the interpretation of how the function behaves.