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The first derivative of a function is a powerful tool in calculus that helps us understand how the function behaves at any given point. It essentially tells us the slope of the function at each point. For the function given, we need to determine how it increases or decreases over its domain. To start, we compute the first derivative of the function \( f(x) = 5 + 20x - x^5 \) using the power rule. This results in the derivative \[ f'(x) = 20 - 5x^4. \]
This derivative expression will help us find critical points by setting it to zero and solving for \( x \). These points, where the first derivative equals zero, usually indicate where the function changes direction either from increasing to decreasing or vice versa.
In our particular function, setting \( 20 - 5x^4 = 0 \) allows us to solve and find the critical points, which are the values of \( x \) such that \( x = \sqrt{2} \) and \( x = -\sqrt{2} \). These points are crucial as they divide the function into intervals where behavior changes. We will explore these intervals in the next section.

Determining where a function is increasing or decreasing is straightforward once we have the first derivative. When\( f'(x) > 0 \), the function is increasing, and when \( f'(x) < 0 \), it's decreasing.
To identify these intervals for \( f(x) = 5 + 20x - x^5 \), we examine the sign of \( f'(x) = 20 - 5x^4 \) in the critical point-divided regions: \((-fty, -\sqrt{2})\), \((-\sqrt{2}, \sqrt{2})\), and \((\sqrt{2}, fty)\).

• For \(x = -2\), a test point in \((-fty, -\sqrt{2})\), \( f'(-2) = -60 \), showing a decreasing trend.
• For \(x = 0\), in \((-\sqrt{2}, \sqrt{2})\), \( f'(0) = 20 \), indicating an increasing trend.
• For \(x = 2\), in \((\sqrt{2}, fty)\), \( f'(2) = -60 \), showing another decrease.

This means the function decreases to \( x = -\sqrt{2} \), then increases up to \( x = \sqrt{2} \), and finally decreases again beyond this point. Identifying these sequences helps us predict how the function behaves graphically and aids in comprehending changes in trends.

Relative maxima and minima refer to points on a function where it reaches a peak (maximum) or a trough (minimum) in their neighborhoods. These are discovered by analyzing where the function shifts from increasing to decreasing and vice versa.
Based on the function \( f(x) = 5 + 20x - x^5 \) and the critical points \( x = \sqrt{2} \) and \( x = -\sqrt{2} \), we gain insight:

• At \( x = -\sqrt{2} \), the function transitions from decreasing to increasing, indicating a **relative minimum**.
• At \( x = \sqrt{2} \), the function switches from increasing to decreasing, signaling a **relative maximum**.

Such points are pivotal because they offer locations of interest on the function curve where significant changes occur.
Thus, these relative extrema shape our understanding of the overall behavior of the function, and paired with the intervals of increase and decrease, they provide a comprehensive depiction of the graph's form. Understanding these concepts allows for skillful problem-solving and analysis of real-world scenarios modeled by functions.