Problem 51 - Calculus Early Transcendentals: Pearson New International Edition

Question

In each case, is it possible for a function \(F\) with two continuous derivatives to satisfy the following properties? If so sketch such a function. If not, justify your answer. (a) \(F^{\prime}(x)>0, F^{\prime \prime}(x)>0\), while \(F(x)<0\) for all \(x\). (b) \(F^{\prime \prime}(x)<0\), while \(F(x)>0\). (c) \(F^{\prime \prime}(x)<0\), while \(F^{\prime}(x)>0\).

Step-by-Step Solution

Answer

(a) Not possible; (b) Possible; (c) Possible.

Step 1: Assess Condition (a)

We need to determine if a function \( F \) can satisfy both \( F'(x) > 0 \) and \( F''(x) > 0 \), while \( F(x) < 0 \) for all \( x \). For \( F'(x) > 0 \), the function is monotonically increasing, and for \( F''(x) > 0 \), the curve is concave up, which means the rate of increase is itself increasing. However, if \( F(x) < 0 \) for all \( x \), this implies the function remains below the \( x \)-axis, which contradicts the nature of \( F''(x) > 0 \) over any interval, as eventually the increasing function should cross the \( x \)-axis. Therefore, it's not possible for such a function to exist.

Step 2: Assess Condition (b)

We consider if there exists a function \( F \) such that \( F(x) > 0 \) for all \( x \), and \( F''(x) < 0 \). The condition \( F''(x) < 0 \) indicates that the function is concave down or the rate of increase is decreasing. A simple example would be a diminishing positive function: \( F(x) = -x^2 + 1 \) over an appropriate domain (e.g., near 0). It demonstrates \( F(x) \) is positive, yet the curve itself initial increases and then starts to decrease consistently within its domain limits. Thus, such a function can exist.

Step 3: Assess Condition (c)

We examine the possibility of having \( F'(x) > 0 \) and \( F''(x) < 0 \) for a function \( F \). \( F'(x) > 0 \) indicates the function is increasing, whereas \( F''(x) < 0 \) suggests the rate of increase is decreasing. This is feasible, as seen in functions like \( F(x) = \sqrt{x} \) or \( F(x) = \log(x) \) (for positive \( x \)). These functions show a decreasing slope but continue to increase. Therefore, it is possible for such a function to exist.

Key Concepts

Derivative

The derivative of a function, often denoted as \( F'(x) \), is a fundamental concept in calculus. It represents the rate at which the function value changes as its input changes. In simpler terms, it's like the speed of the function at any given point. If \( F'(x) > 0 \), it means the function is increasing—a positive slope. Conversely, if \( F'(x) < 0 \), the function is decreasing, exhibiting a negative slope.

• Positive Derivative (Increasing Function): If \( F'(x) > 0 \), as you move along the x-axis, the function's y-values (or heights on a graph) keep climbing upward.
• Negative Derivative (Decreasing Function): On the other hand, if \( F'(x) < 0 \), the y-values descend, indicating a drop as you traverse the x-axis.

Derivatives provide critical insight into the behavior and direction of a function.

Concavity

Concavity describes the curvature of a function and is determined by its second derivative, \( F''(x) \). The concavity tells us how the slope of a function changes: either speeding up or slowing down.

• Concave Up (\( F''(x) > 0 \)): This indicates that the slope of the function is increasing. The graph shapes upward like a bowl. Any point on such a curve will promote acceleration in its rate of increase.
• Concave Down (\( F''(x) < 0 \)): In this scenario, the slope of the function is decreasing. The graph resembles an upside-down bowl. Points here experience a reduction in the rate of increase or speed.

Understanding concavity aids in predicting how a function might grow or shrink at various intervals.

Monotonic Function

A monotonic function maintains constant deviation in its direction, either always going up or down but never reversing. When \( F'(x) > 0 \), the function is monotonically increasing, meaning its values never fall as the independent variable grows.

• Monotonic Increase: Every increase in \( x \) results in an equal or greater increase in \( F(x) \). This is often visualized as a consistent uphill motion on a graph.
• Monotonic Decrease: Conversely, with \( F'(x) < 0 \), the function is always falling, showing a consistent downhill trend.

Monotonic functions help simplify calculations and analysis, ensuring predictability in a function's path.

Function Behavior

The behavior of a function is encompassed by how it grows, shrinks, bends, speeds up, or slows down. Essential to this analysis are its derivatives, which provide a narrative of how the function behaves over time:

• **Growth and Decline:** A combination of \( F'(x) \) and \( F''(x) \) illustrates not just the rise or fall of a function but how aggressively it performs these actions.
• **Changes in Speed:** While \( F'(x) \) gauges speed, \( F''(x) \) critiques the change in speed—the acceleration or deceleration expected in its trajectory.
• **Graphical Representation:** By observing graphs stemming from these derivations, predictions about eventual intersections with axes or potential peaks and troughs become clearer.

Through these lenses, one can thoroughly explore and forecast the intricate dances of mathematical functions.