Problem 11 - University Calculus: Early Transcendentals

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: \(-2, 0\). Decreasing on \((-\infty, -2)\), Increasing on \((-2, 0)\) and \((0, \infty)\). Local minimum at \(-2\).

Step 1: Find Critical Points

Critical points of the function occur where the derivative is zero or undefined. Set \(f'(x) = x^{-1/3}(x+2) = 0\). For this product to be zero, either \(x^{-1/3} = 0\) or \(x+2 = 0\). Since \(x^{-1/3} = 0\) is never true, we solve \(x+2 = 0\), obtaining \(x = -2\). Also, consider where the derivative is undefined, which occurs at \(x = 0\) due to \(x^{-1/3}\). Therefore, the critical points are \(x = -2\) and \(x = 0\).

Step 2: Determine Increasing/Decreasing Intervals

Analyze the sign of \(f'(x)\) to find where \(f\) is increasing or decreasing. Consider the test intervals \((-\infty, -2), (-2, 0), (0, \infty)\):- For \(x \in (-\infty, -2)\), choose \(x = -3\): \(f'(-3) = (-3)^{-1/3}(-3+2) < 0\), so \(f\) is decreasing.- For \(x \in (-2, 0)\), choose \(x = -1\): \(f'(-1) = (-1)^{-1/3}(1) > 0\), so \(f\) is increasing.- For \(x \in (0, \infty)\), choose \(x = 1\): \(f'(1) = 1^{-1/3}(3) > 0\), so \(f\) is increasing.

Step 3: Find Local Maximum/Minimum Values

Using the information from previous steps, analyze where the function transitions between increasing and decreasing behavior to determine local extrema:- At \(x = -2\), the function transitions from decreasing (left) to increasing (right), indicating a local minimum at \(x = -2\).- At \(x = 0\), the function transitions from increasing (left) to increasing (right), so \(x = 0\) is neither a maximum nor a minimum.

Key Concepts

Critical Points

In calculus, finding critical points is essential to understand the behavior of a function. Critical points occur where the derivative of the function is either zero or undefined. They are potential locations for local maxima or minima. This is because they represent points where the rate of change of the function levels out, which can hint at peaks, valleys, or flat sections.

To find the critical points of a given function derivative, say, the derivative is expressed as \(f'(x) = x^{-1/3} (x + 2)\), you must set \(f'(x) = 0\). For our function, this zero happens when the term \(x + 2 = 0\), giving us \(x = -2\).

But that's not all. We must also inspect where the derivative might be undefined. For \(x^{-1/3}\), the derivative is not defined at \(x = 0\), so \(x = 0\) is another critical point. Ultimately, the critical points of this function are \(x = -2\) and \(x = 0\). These points help us determine intervals of increase or decrease and locate any local extrema.

Increasing/Decreasing Functions

Functions can have different behaviors in various intervals, and understanding where a function increases or decreases is crucial. A function is increasing if its derivative is positive over an interval, meaning the function's value rises as you move from left to right across that interval.

Oppositely, it's decreasing if the derivative is negative, meaning the function's value falls. With our function derivative \(f'(x) = x^{-1/3}(x + 2)\), test intervals around the critical points can be analyzed for positivity or negativity.

Consider the intervals based on the critical points we've found:

• (\(-\infty, -2\)): Choose a test value, say \(x = -3\). Substituting, we find that \(f'(-3) < 0\), indicating the function decreases in this interval.
• (\(-2, 0\)): Pick a test value, \(x = -1\). Here \(f'(-1) > 0\), showing the function increases.
• (\(0, \infty\)): A test value \(x = 1\) gives \(f'(1) > 0\), indicating the function is likewise increasing.

These insights assist in visualizing how the function behaves over its domain.

Local Maximum and Minimum Values

Local maximum and minimum values, often called local extrema, provide crucial insights into a function's graph. These extrema occur where a function transitions its behavior between increasing and decreasing patterns. When examining critical points, consider the change in the function's increase or decrease to identify these points.

With the function we are analyzing, let's observe the critical points we found:

• At \(x = -2\), the function goes from decreasing to increasing, indicating a local minimum at this point. The progression hints that it is a valley on the graph.
• On the other hand, at \(x = 0\), because the function increases on both sides of this point, it establishes neither a peak nor a valley. Hence, \(x = 0\) isn't a local maximum or minimum.

Understanding where these occurrences happen helps sketch and comprehend the overall structure of the function.