Problem 11 - Thomas Calculus

Question

Answer the following questions about the functions whose derivatives are given. a. What are the critical points of \(f ?\) b. On what open intervals is \(f\) increasing or decreasing? c. At what points, if any, does \(f\) assume local maximum and minimum values? $$ f^{\prime}(x)=x^{-1 / 3}(x+2) $$

Step-by-Step Solution

Answer

Critical points: \( x = -2, 0 \); Increasing on \((-\infty, -2) \cup (0, \infty)\); Decreasing on \((-2, 0)\); Local max at \( x = -2 \); Local min at \( x = 0 \).

Step 1: Find Critical Points

To determine the critical points of \( f \), we need to find where the derivative \( f'(x) = x^{-1/3}(x+2) \) is equal to zero or undefined. Start by setting \( f'(x) = 0 \):\[ x^{-1/3} (x + 2) = 0 \]This implies either \( x = 0 \) or \( x + 2 = 0 \).The second condition gives \( x = -2 \).\( f'(x) \) is undefined when \( x^{-1/3} \) is undefined, which is at \( x = 0 \).Thus, the critical points are \( x = -2 \) and \( x = 0 \).

Step 2: Determine Increasing and Decreasing Intervals

To find where \( f \) is increasing or decreasing, test the sign of \( f'(x) = x^{-1/3}(x+2) \) on intervals determined by the critical points (-∞, -2), (-2, 0), and (0, ∞).- For \( x \in (-\infty, -2) \), choose \( x = -3 \): \( f'(-3) = (-3)^{-1/3}(-3 + 2) \) is positive.- For \( x \in (-2, 0) \), choose \( x = -1 \): \( f'(-1) = (-1)^{-1/3}(-1 + 2) \) is negative.- For \( x \in (0, \infty) \), choose \( x = 1 \): \( f'(1) = 1^{-1/3}(1 + 2) \) is positive.Therefore, \( f \) is increasing on \((-\infty, -2) \cup (0, \infty)\) and decreasing on \((-2, 0)\).

Step 3: Identify Local Maxima and Minima

To find the local maxima and minima, investigate the behavior of \( f \) around the critical points \( x = -2 \) and \( x = 0 \).- At \( x = -2 \), \( f \) changes from increasing to decreasing, indicating a local maximum.- At \( x = 0 \), \( f \) changes from decreasing to increasing, indicating a local minimum.Thus, \( f \) has a local maximum at \( x = -2 \) and a local minimum at \( x = 0 \).

Key Concepts

Increasing and Decreasing Intervals

In the world of calculus, knowing where a function is increasing or decreasing helps us understand its behavior over different intervals. This is essential to predict how it changes and develops naturally.
When analyzing a function's derivative like \( f'(x) = x^{-1/3}(x+2) \), you can determine where the function \( f \) increases or decreases by assessing the sign of \( f'(x) \) over specific intervals.

To find these intervals, first identify the critical points where \( f'(x) = 0 \) or is undefined. In this scenario, the critical points are \( x = -2 \) and \( x = 0 \).
Then, test values from the intervals formed by these points:

• For \( x \in (-\infty, -2) \), since \( f'(-3) \) is positive, \( f \) increases.
• For \( x \in (-2, 0) \), since \( f'(-1) \) is negative, \( f \) decreases.
• For \( x \in (0, \infty) \), since \( f'(1) \) is positive, \( f \) increases.

This method reveals that \( f \) is increasing on \( (-\infty, -2) \cup (0, \infty) \), and decreasing on \( (-2, 0) \). Understanding these intervals provides insight into how the function's output behaves across its domain.

Local Maxima and Minima

The concepts of local maxima and minima are crucial for understanding the high and low points of a function. These critical points refer to positions where the function takes on local extreme values—either a peak or a trough.
In our exercise, analyzing \( f'(x) = x^{-1/3}(x+2) \) helps us identify local extrema.

A local maximum occurs where \( f \) changes from increasing to decreasing. For this function, observe that:

• At \( x = -2 \), since \( f \) transitions from increasing to decreasing, there is a local maximum.

Conversely, a local minimum arises where \( f \) transitions from decreasing to increasing.

• At \( x = 0 \), since \( f \) goes from decreasing to increasing, a local minimum exists.

Thus, this derivative analysis informs us that the function \( f \) is at a local maximum at \( x = -2 \) and at a local minimum at \( x = 0 \). Understanding these critical points is significant for sketching graphs and predicting the function's behavior near those regions.

Derivative Analysis

Derivatives play a pivotal role in calculus by providing a tool to explore the rate of change of functions. Through derivative analysis, we gain a comprehensive view of where and how a function behaves differently.
For the function with the derivative \( f'(x) = x^{-1/3}(x+2) \), derivative analysis illuminates key information about its slope and behavior. This involves finding critical points, which we identified as \( x = -2 \) and \( x = 0 \). The critical points allow us to break the function's domain into intervals where it behaves uniquely.

Check the behavior of \( f'(x) \):

• If \( f'(x) > 0 \), the function is increasing on that interval.
• If \( f'(x) < 0 \), the function decreases on that interval.

By drafting values in these intervals and calculating signs, derivative analysis summarizes the function's attributes over its domain. Understanding this concept helps students not only in plotting curves but also in mastering more complex calculus ideas such as optimization problems and concavity analysis.