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A relative maximum of a function occurs at a point where the function's value is higher than at any nearby points. It is not necessarily the highest point of the whole function, but it's a 'peak' in its local neighborhood. Find these points using the first-derivative test.

The first-derivative test is an efficient method to find relative maxima and minima:

• First, find the function's first derivative, denoted as \( f'(x) \). This step will help you understand how the function is changing at every point.
• Next, set \( f'(x) = 0 \). Solve for \( x \) to find critical points where the function could potentially reach a relative maximum or minimum.
• Now, test intervals around each critical point to determine whether the first derivative is positive or negative.
• If the sign of \( f'(x) \) changes from positive to negative around a critical point, it means the function is rising then falling, indicating a relative maximum at that point.

In the provided exercise with the function \( f(x) = x^3 - 6x^2 + 1 \), the critical point at \( x = 0 \) has \( f'(x) \) changing from positive to negative, showing a relative maximum at this point.
Finally, calculate the function value at \( x = 0 \) to get the maximum value, which was \( f(0) = 1 \). So, the relative maximum is at \( (0, 1) \).

A relative minimum of a function happens at a point where the function's value is lower than at any nearby points. It's a 'valley' in its local area but not necessarily the lowest point of the entire function.

To determine a relative minimum, follow these steps with the first-derivative test:

• Find the first derivative of the function \( f'(x) \), which tells you where the slopes of the function change.
• Set the first derivative \( f'(x) = 0 \) to locate critical points.
• Check the intervals around each critical point to see if the sign of \( f'(x) \) switches from negative to positive.
• If the sign of \( f'(x) \) changes from negative to positive, this means the function is falling then rising, indicating a relative minimum.

For instance, in the exercise with function \( f(x) = x^3 - 6x^2 + 1 \), the critical point at \( x = 4 \) has its first derivative changing from negative to positive, implying a relative minimum.Lastly, calculate the function value at \( x = 4 \), giving \( f(4) = -31 \). Therefore, the relative minimum is at \( (4, -31) \).

Calculus is a branch of mathematics focusing on changes and motion. It mainly involves two major concepts: derivatives and integrals. Derivatives help in understanding the rate of change, while integrals help in finding the total accumulation or area under curves.

• The derivative of a function, \( f'(x) \), represents the slope or the rate of change of the function at any given point. It shows how the function’s value is changing at each point.

In the given exercise, we use the first derivative to determine critical points where the function could have relative maxima or minima. Following this, the first-derivative test helps in analyzing these points to deduce whether the function is reaching a peak (relative maximum) or a trough (relative minimum).

• Understanding calculus and using tools like the first-derivative test provides a systematic way to analyze functions thoroughly. It’s essential in many scientific disciplines and practical applications.

Conclusively, calculus and its derivative concepts are fundamental to understanding changes in functions and solving problems related to relative maximum and minimum points, as seen in the provided example.