Question

In Exercises , identify the point of diminishing returns for the inputoutput function. For each function, \(R\) is the revenue and \(x\) is the amount spent on advertising. Use a graphing utility to verify your results. $$ R=-\frac{4}{5}\left(x^{3}-9 x^{2}-27\right), \quad 0 \leq x \leq 5 $$

Short answer

The point of diminishing returns for the given function is \(x=3\).

Step-by-step solution

Calculating the first derivative

Determine the first derivative of the function \(R\) with respect to \(x\). It is \(-3x^{2}+\frac{36x}{5}\).

Calculating the second derivative

Determine the second derivative of the function \(R\) with respect to \(x\). It is \(-6x + \frac{36}{5}\).

Finding the point where second derivative is negative

The point of diminishing returns is the first point where the second derivative is less than zero. To find this point, we solve the inequality \(-6x+\frac{36}{5}<0\). Solving gives \(x>3\). However, because of the limitations given in the problem \((0 \leq x \leq 5)\), the point of diminishing returns is when \(x=3\) within our limitations.

Verifying via graphing utility

To confirm this result, graph the function within its given limits (0 to 5) using a graphing utility. Observe that the slope of the curve starts to decrease for \(x\) values larger than 3, confirming that 3 is the point of diminishing returns within the given limits.

Key concepts

Revenue Function

In this exercise, we explore the concept of a revenue function, which is a mathematical expression representing the revenue generated based on certain input variables. In our example, we consider the revenue function: \(-\frac{4}{5}(x^{3}-9x^{2}-27) \) where \( R \) represents the revenue and \( x \) denotes the amount spent on advertising.

This function lets businesses or individuals predict how changes in advertising expenditure can impact their revenue. Specifically, in this exercise, \( x \) is bounded between 0 and 5, indicating the range of investment that should be analyzed.

A deeper understanding of this function is crucial as it serves as the basis for determining key financial insights like the point of diminishing returns.

First Derivative

The first derivative of a function provides insights about the rate at which the function's value changes with respect to changes in its variable, in this case, \( x \). By calculating the first derivative of our revenue function, we derive:

• \( R' = -3x^{2} + \frac{36x}{5} \)

This first derivative tells us how the revenue changes as we adjust our advertising expenditure. A positive derivative indicates increasing revenue, while a negative derivative suggests that revenue is decreasing.

Analyzing these changes helps identify critical points, such as local maxima or minima, which can be crucial in understanding performance gains or losses.

In simpler terms, the first derivative acts as a guiding tool to understand the efficiency of ad spending and its impact on revenue.

Second Derivative

The second derivative provides insights into the concavity of the function, which helps determine the point of diminishing returns. For this revenue function, the second derivative is:

• \( R'' = -6x + \frac{36}{5} \)

By solving the inequality \( R'' < 0 \), we can identify the point of diminishing returns in the context of our given input range (0 to 5). The inequality \(-6x + \frac{36}{5} < 0 \) indicates where the function changes from concave up to concave down, suggesting a decrease in the rate of revenue gain per additional dollar spent on advertising.

This translates to the insight that spending beyond \( x = 3 \) leads to less efficient revenue generation. Understanding this turning point helps in strategizing optimal allocation of advertising budgets.

Graphing Utility

Visualizing mathematical concepts aids in comprehension, and this is where a graphing utility becomes exceptionally useful. By graphically representing the revenue function, one can observe the behavior of the curve over the specified range for \( x \), from 0 to 5.

When graphed, notice how the slope of the revenue curve starts to decrease after \( x = 3 \). This visual cue aligns with our prior analytical findings using the second derivative, validating \( x = 3 \) as the point of diminishing returns.

Graphing not only serves as a verification tool but also enriches understanding, making it tangible by showing how theoretical concepts play out in practice. Overall, engaging with graphing utilities empowers students to link abstract calculation steps with real-world fiscal decision-making processes.