Problem 70 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, use a graphing utility to graph \(f, f^{\prime}\). and \(f^{\prime \prime}\) in the same viewing window. Graphically locate the relative extrema and points of inflection of the graph of \(f\). State the relationship between the behavior of \(f\) and the signs of \(f^{\prime}\) and \(f^{\prime \prime}\) $$ f(x)=-\frac{1}{20} x^{5}-\frac{1}{12} x^{2}-\frac{1}{3} x+1, \quad[-2,2] $$

Step-by-Step Solution

Answer

By graphing \(f\), \(f'\), and \(f''\), we can visualize the behavior of \(f\) and determine its relative extrema and points of inflection. The positive or negative signs of \(f'\) and \(f''\) corroborate this analysis.

Step 1: Graph function and its derivatives

Use a graphing utility to plot the function \(f(x)=-\frac{1}{20} x^{5}-\frac{1}{12} x^{2}-\frac{1}{3} x+1\), and its first and second derivatives in the same viewing window. The first derivative of a function expresses its slope or rate of change, and the second derivative represents the rate of change of the rate of change, which translates to the acceleration or curvature. Check for different x values within the given interval [-2,2] and mark the graph.

Step 2: Identify relative extrema

Observe the graph of the function \(f\). If there is a point on the graph where \(f\) goes from increasing to decreasing, that point is a relative maximum. If the graph changes from decreasing to increasing, that point is a relative minimum. By checking the graph, we can find these relative extrema.

Step 3: Identify points of inflection

An inflection point is a point on the graph where the graph changes its concavity. In other words, it goes from being concave up (shaped like a U) to concave down (shaped like an ∩), or vice versa. We locate these points by observing the changes in the graph of the second derivative.

Step 4: State the relationship

After identifying the relative extrema and points of inflection, we can state the relationship between the signs of \(f'\) and \(f''\) and the behavior of \(f\). When \(f'\) is positive, \(f\) is increasing, and when \(f'\) is negative, \(f\) is decreasing. Additionally, when \(f''\) is positive, \(f\) is concave up, and when \(f''\) is negative, \(f\) is concave down.

Key Concepts

Derivatives

Understanding derivatives is crucial for analyzing the behavior of functions like the one in the given exercise. A derivative measures the rate at which a function is changing at any given point. For example, the first derivative, denoted as \( f' \), tells us how the slope of the original function \( f(x) \) changes. By calculating the first derivative, we know if the function is increasing or decreasing at different points. An increasing function has a positive \( f' \), while a decreasing function has a negative \( f' \).

The second derivative, \( f'' \), provides additional insight by showing how the rate of change itself is changing. You can think of it as representing the "acceleration" of \( f(x) \). This derivative helps us understand the concavity or the "bend" of the graph. Here's why derivatives are helpful:

• **Identifying behavior:** Positive \( f' \) suggests an upward trend, while negative \( f' \) suggests a downward trend.
• **Finding curvature:** Positive \( f'' \) indicates the graph is curving upwards, like a smile, and negative \( f'' \) indicates it is curving downwards, like a frown.

By graphing \( f \), \( f' \), and \( f'' \) with a graphing utility, we can visualize these changes and better understand the function's behavior in the interval \([-2, 2]\).

Relative Extrema

The relative extrema of a function, including relative maxima and minima, are points where the function reaches a local maximum or minimum compared to the nearby values. These points are significant in calculus because they often represent important values in optimization problems. Identifying these points involves examining the first derivative \( f' \).

• **Relative maximum:** Occurs where \( f' \) changes from positive to negative. This indicates a point where the function switches from increasing to decreasing.
• **Relative minimum:** Occurs where \( f' \) changes from negative to positive. This suggests a transition from decreasing to increasing.

To find these points in the function \( f(x) \), use the graphing utility to plot \( f' \) and look for places where the derivative crosses the x-axis. These are your critical points where \( f' = 0 \). Then, assess the sign change of \( f' \) around these points to determine if they are maxima or minima.

Points of Inflection

Points of inflection on a graph are fascinating because they show where the function changes its concavity. In other words, these points signal a change from being concave up (curving upwards) to concave down (curving downwards), or vice versa. To find these points, we need to focus on the second derivative, \( f'' \).

• **Concave up to concave down:** If \( f'' \) changes from positive to negative, the function shifts from a concave up shape to concave down. Think of it as changing from a bowl shape to an arch shape.
• **Concave down to concave up:** Conversely, if \( f'' \) changes from negative to positive, the function goes from concave down to concave up.

Using a graphing utility, plot \( f'' \) and observe where it crosses the x-axis. These crossings are where \( f'' = 0 \), indicating potential points of inflection. Checking the sign change of \( f'' \) around these zero crossings will confirm the presence of an inflection point.