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In mathematics, relative extrema refer to the peaks and valleys in a graph of a function. These points are critical because they represent where the function changes direction. A relative maximum is a point where the graph reaches a peak, and as such, it's higher than all nearby points. A relative minimum, on the other hand, is a trough, where the graph reaches a low point relative to surrounding values.

For example, in the exercise you're working on, there's a relative maximum at \(x = -1\) and a relative minimum at \(x = 3\). This means the function increases to \(x = -1\) and decreases afterwards, reaching a high. Similarly, it decreases to \(x = 3\), where it then starts to rise again. Understanding these points helps us sketch and interpret the overall shape of the function, marking significant transition points.

• A relative maximum at \(x = -1\) means the function peaks here.
• A relative minimum at \(x = 3\) indicates a valley.

This makes the functions alternate between increasing and decreasing trends around these points.

Critical points are where the first derivative of a function equals zero or is undefined, indicating potential changes in the function’s increasing or decreasing behavior. These points are crucial because they often signal where a function might have relative maximums or minimums.

In the given problem, we identified critical points at \(x = -1\) and \(x = 3\). This is because the first derivative \(f'(x)\) is equal to zero at these points. Critical points don't always mean a change in direction, but they are the only places where a function can change from increasing to decreasing or vice versa.

• At \(x = -1\) and \(x = 3\), the derivative \(f'(x)\) is zero.
• This means these are locations where the function's rate of change might shift sign.

In this exercise, these points are also the locations of relative extrema.

The first derivative test is a valuable tool to determine whether a critical point is a relative maximum, a relative minimum, or neither. It involves analyzing the sign of the first derivative before and after the critical point.

Here's a simple way to conduct the test:

• If \(f'(x)\) changes from positive to negative at a critical point, the point is a relative maximum.
• If \(f'(x)\) changes from negative to positive, it's a relative minimum.
• If there's no sign change, the point is neither.

In the exercise, for \(x = -1\), \(f'(x)\) changes from positive to negative, indicating a relative maximum. Conversely, at \(x = 3\), \(f'(x)\) changes from negative to positive, marking a relative minimum. This aligns perfectly with the problem's statement and helps you verify the behavior of the function around these points.

Understanding where a function is increasing or decreasing is essential for fully grasping its shape and behavior. A function is increasing on intervals where its derivative is positive and decreasing where its derivative is negative.

In the given problem, the intervals are:

• For \(x < -1\) and \(x > 3\), \(f'(x) > 0\), meaning \(f\) is increasing.
• For \(-1 < x < 3\), \(f'(x) < 0\), meaning \(f\) is decreasing.

This tells us a lot about the function's path:

• Before \(x = -1\), the function climbs towards a peak at \(x = -1\).
• Then it falls as it moves towards \(x = 3\), hitting a trough.
• After \(x = 3\), the function climbs again.

Recognizing these intervals is crucial when sketching or examining the function graph. It helps to anticipate where the function rises or descends, ensuring the curves are correctly oriented in relation to critical points and relative extrema.