Problem 74 - Calculus of a Single Variable

Question

Think About It In Exercises 73 and \(74,\) the function \(f\) is differentiable on the indicated interval. The table shows \(f^{\prime}(x)\) for selected values of \(x\) . (a) Sketch the graph of \(f,\) approximate the critical numbers, and (c) identify the relative extrema. \(f\) is differentiable on \([0, \pi]\) $$ \begin{array}{|c|c|c|c|c|}\hline x & {0} & {\pi / 6} & {\pi / 4} & {\pi / 3} & {\pi / 2} \\ \hline f^{\prime}(x) & {3.14} & {-0.23} & {-2.45} & {-3.11} & {0.69} \\ \hline x & {2 \pi / 3} & {3 \pi / 4} & {5 \pi / 6} & {\pi} \\\ \hline f^{\prime(x)} & {3.00} & {1.37} & {-1.14} & {-2.84} \\\ \hline\end{array} $$

Step-by-Step Solution

Answer

Approximate critical numbers are: \(\pi / 6\), \(2 \pi / 3\), and \(5 \pi / 6\). There are local maxima at \(x = \pi / 6\) and \(x = 5 \pi / 6\) and a local minimum at \(x = 2 \pi / 3\). Sketch of the graph shows the function increasing and decreasing at these intervals accurately.

Step 1: Identify intervals where function is increasing and decreasing

To determine where the function is increasing or decreasing, we look at the sign of the derivative \(f'(x)\). If \(f'(x) > 0\), \(f\) is increasing. If \(f'(x) < 0\), \(f\) is decreasing. From the table:\n- \(f\) is increasing on the intervals: \(x = [0, \pi / 6)]\) and \([2 \pi / 3, 5 \pi / 6]\)\n- \(f\) is decreasing on the intervals: \(x = (\pi / 6, 2 \pi / 3)\) and \([5 \pi / 6, \pi]\)

Step 2: Determine critical numbers

Critical numbers occur where the derivative is equal to 0 or undefined. From the table, we see that the derivative doesn't equal 0 or undefined, thus we examine the x-values where the sign of the derivative changes. They are \(\pi / 6\), \(2 \pi / 3\), and \(5 \pi / 6\). These are the critical numbers.

Step 3: Identify relative extrema

The relative extrema occur at the critical numbers. If the function changes from increasing to decreasing at a critical number, then the function has a local maximum there. If it changes from decreasing to increasing, it has a local minimum. So, the function \(f\) has:\n- A local maximum at \(x = \pi / 6\) (increasing to decreasing)\n- A local minimum at \(x = 2 \pi / 3\) (decreasing to increasing)\n- A local maximum at \(x = 5 \pi / 6\) (increasing to decreasing)

Step 4: Sketch the graph

In our sketch, we should show:\n- The function increasing in the intervals \(x = [0, \pi / 6)]\) and \([2 \pi / 3, 5 \pi / 6]\)\n- Decreasing in the intervals \(x = (\pi / 6, 2 \pi / 3)\) and \([5 \pi / 6, \pi]\)\n- Local maxima at \(x = \pi / 6\) and \(x = 5 \pi / 6\)\n- A local minimum at \(x = 2 \pi / 3\)\nNote: Drawing the actual shape of the function \(f\) is beyond scope since we don't have information on the original function.

Key Concepts

Critical Numbers

In the world of differentiable functions, **critical numbers** play an important role. They help us identify where a function has potential peaks and valleys. Critical numbers are the x-values at which the derivative of a function is either zero or undefined. These points indicate that the function's rate of change is either pausing or experiencing a jump.

In our specific example, the derivative does not produce any undefined values, and the table doesn't show any zeros. Instead, we observe changes in the sign of the derivative at specific points. These locations are still crucial since they indicate where the function's behavior alters. In this exercise, critical numbers are at \( \frac{\pi}{6}, \frac{2\pi}{3}, \text{and} \frac{5\pi}{6} \). At these points, there is a transition between increasing and decreasing behaviors in the function.

Relative Extrema

Finding **relative extrema** means locating the high and low points of a function within a certain interval. These extrema signify local peaks and troughs, which are key to understanding the function's overall behavior. These occur at critical numbers when there is a change in the derivative's sign.

In the exercise, we observe:

• A transition from increasing to decreasing at \( x = \frac{\pi}{6} \), indicating a local maximum.
• A shift from decreasing to increasing at \( x = \frac{2\pi}{3} \), signifying a local minimum.
• Another change from increasing to decreasing at \( x = \frac{5\pi}{6} \), implying a local maximum.

Recognizing these shifts helps in sketching the function and understanding its nature better.

Increasing and Decreasing Intervals

To determine where functions are moving upwards or downwards, we inspect the derivative's sign. If a function's derivative \( f'(x) \) is positive, the function is in an **increasing interval**; if negative, it is a **decreasing interval**. Observing these intervals provides insight into the function's trajectory.

For the given table:

• Increasing intervals are from the start \( x = [0, \frac{\pi}{6}] \) and between \( [\frac{2\pi}{3}, \frac{5\pi}{6}] \), where the derivative is positive.
• Decreasing intervals occur from \( x = (\frac{\pi}{6}, \frac{2\pi}{3}) \) and on \( [\frac{5\pi}{6}, \pi] \), as the derivative takes on negative values.

These intervals are a roadmap of how the function grows and declines.

First Derivative Test

The **first derivative test** is a method to identify the nature of critical points, providing a clear picture of whether they are relative maxima, minima, or neither. By examining the sign changes of \( f'(x) \) at critical numbers, one can distinguish the kind of extrema present.

In our problem, applying this test reveals:

• At \( x = \frac{\pi}{6} \), a change from positive to negative in \( f'(x) \) establishes a local maximum.
• At \( x = \frac{2\pi}{3} \), the derivative switches from negative to positive, confirming a local minimum.
• Lastly, at \( x = \frac{5\pi}{6} \), the derivative shifts back from positive to negative, indicating another local maximum.

This test solidifies our understanding of how the function behaves around its critical numbers.