Problem 27 - Calculus Early Transcendentals

Question

Use the given derivative to find all critical points of \(f\) and at each critical point determine whether a relative maximum, relative minimum, or neither occurs. Assume in each case that \(f\) is continuous everywhere. $$ f^{\prime}(x)=\frac{2-3 x}{\sqrt[3]{x+2}} $$

Step-by-Step Solution

Answer

Relative minimum at \(x = -2\); relative maximum at \(x = \frac{2}{3}\).

Step 1: Find where the derivative is zero or undefined

To find the critical points, we need to solve for when the first derivative, \(f'(x) = \frac{2 - 3x}{\sqrt[3]{x+2}}\), is zero or undefined. Start by setting the numerator equal to zero: \(2 - 3x = 0\). Solve for \(x\) to get \(x = \frac{2}{3}\). Next, find where the derivative might be undefined by setting the denominator to zero: \(\sqrt[3]{x + 2} = 0\). Solving this gives \(x = -2\). So, the critical points are \(x = \frac{2}{3}\) and \(x = -2\).

Step 2: Determine the nature of critical points using the First Derivative Test

For the First Derivative Test, evaluate the function's behavior around the critical points. Examine intervals around these points: \((-\infty, -2), (-2, \frac{2}{3}), (\frac{2}{3}, \infty)\).- For \(x < -2\), choose \(x = -3\): \(f'(-3) = \frac{2-3(-3)}{\sqrt[3]{-3+2}} = \frac{11}{\sqrt[3]{-1}} = -11\), which means \(f'\) is negative.- For \(-2 < x < \frac{2}{3}\), choose \(x = 0\): \(f'(0) = \frac{2-3(0)}{\sqrt[3]{0+2}} = \frac{2}{\sqrt[3]{2}}\), which is positive.- For \(x > \frac{2}{3}\), choose \(x = 1\): \(f'(1) = \frac{2-3(1)}{\sqrt[3]{1+2}} = -1\), which is negative.At \(x = -2\), \(f'\) changes from negative to positive, indicating a relative minimum. At \(x = \frac{2}{3}\), \(f'\) changes from positive to negative, indicating a relative maximum.

Key Concepts

First Derivative Test

The First Derivative Test is a powerful tool used to determine the nature of critical points of a function. This test leverages changes in the sign of the derivative around critical points to provide insights into whether the function has a relative maximum or minimum. Here's how it works:

• Find the critical points by determining where the derivative is zero or undefined.
• Choose test points in the intervals around each critical point to evaluate the sign of the derivative.
• Observe the change in sign of the first derivative across these intervals.

If the first derivative changes:

• From positive to negative across a critical point, the function has a relative maximum there.
• From negative to positive across a critical point, the function has a relative minimum there.
• It remains the same, then the point is neither a relative maximum nor minimum.

By following these steps, the First Derivative Test provides a straightforward approach to understand the behavior of functions at their critical points.

Relative Maximum

A relative maximum is a point where a function reaches a peak value over a small interval compared to adjacent values. It is not necessarily the highest point in the entire function; it just needs to be higher than the points nearby. To identify a relative maximum using derivatives, you'll want to look for places where the function changes its increasing trend to a decreasing one.

• At a relative maximum, the first derivative changes from positive (increasing function) to negative (decreasing function) as you pass through the critical point.
• This indicates the function was rising as it approached the point and starts falling as it leaves the point.

In the context of the original solution, at the critical point where the derivative sign changes from positive to negative, you can confidently say a relative maximum occurs. This identifies regions where the function peaks relative to its immediate surroundings.

Relative Minimum

Contrary to a relative maximum, a relative minimum is a point where the function reaches a trough or the lowest value over a small interval. Like the relative maximum, it's not necessarily the lowest point across the entire function but is lower than the nearby values. To determine a relative minimum using derivatives:

• Observe where the first derivative shifts from negative to positive as you pass through the critical point.
• Here, the function transitions from a downward trend to an upward trend.

This change implies the function was decreasing before reaching the point and starts increasing just after. In the exercise example, a relative minimum is identified at the critical point where the derivative transitions from negative to positive, indicating a local trough in the function's value at that location.