Question

In Exercises, identify the point of diminishing returns for the input output function. For each function, \(R\) is the revenue and \(x\) is the amount spent on advertising. Use a graphing utility to verify your results. $$ R=\frac{1}{50,000}\left(600 x^{2}-x^{3}\right), \quad 0 \leq x \leq 400 $$

Short answer

The point of diminishing returns for the given revenue function is at \(x = 200\).

Step-by-step solution

Compute First Derivative

Compute the first derivative of the function, \(R'(x)\). This gives the rate of change of revenue with respect to advertising spend. Using the power rule, the first derivative will be \(\frac{1}{50,000} \left(1200x - 3x^{2}\right)\).

Compute Second Derivative

Next, compute the second derivative, \(R''(x)\). This gives the rate of change of the rate of change of revenue (the rate at which the first derivative is changing), and it shows where the function's rate of change is increasing or decreasing. The second derivative will be \(\frac{1}{50,000} (1200 - 6x)\).

Find the Point of Diminishing Returns

Set the second derivative equal to zero and solve for \(x\) to find the point of diminishing returns, as this is where the rate of change of revenue starts decreasing. Solving \(\frac{1}{50,000}(1200-6x) = 0\), we get \(x = 200\) as the point of diminishing returns.

Verification using a Graphing Utility

Finally, plot the function \(R(x)\), and its first and second derivatives on a graphing utility. As observed, the slope of the function starts to decrease after \(x = 200\), affirming that this is indeed the point of diminishing returns.

Key concepts

Input Output Function

An input-output function describes the relationship between inputs into a system and the resulting outputs. In our scenario, the input is the amount spent on advertising, and the output is the revenue generated. The function given is \( R=\frac{1}{50,000}(600x^2-x^3) \).

This function shows how revenue changes as more money is spent on advertising, up to a certain amount. It is essential to understand that not all input increases lead to proportional output increases. The concept of diminishing returns kicks in at a certain point, where additional spending results in lower increases in revenue.

For example, at lower levels of advertising spend, revenue might increase quickly. However, after a certain point, additional spending might yield less increase in revenue. That's the essence of diminishing returns, which can be identified through examining derivatives.

First Derivative

The first derivative of a function, represented as \( R'(x) \), tells us how the revenue changes concerning the change in advertising spending. It provides a measure known as the rate of change. In simple terms, it shows how much additional revenue is generated when advertising spending increases by one unit.

For the given function, the first derivative is \( R'(x) = \frac{1}{50,000} (1200x - 3x^2) \).

This derivative gives a formula to calculate the increase or decrease in revenue for each unit increase in advertising. The sign and value of \( R'(x) \) at any point \( x \) indicate whether the revenue is increasing or decreasing. When the derivative is positive, spending more yields more revenue. However, as \( x \) approaches the point of diminishing returns, this increase slows down.

Second Derivative

The second derivative, expressed as \( R''(x) \), provides insight into the acceleration or deceleration of revenue changes. It calculates the rate at which the rate of change itself is changing.

For our function, \( R''(x) = \frac{1}{50,000} (1200 - 6x) \) helps determine when the revenue growth starts to slow, which highlights the diminishing returns point. When \( R''(x) = 0 \), it shows that the rate of revenue changes is at a maximum or minimum—a critical point.

Solving for this second derivative gives us \( x = 200 \). This point indicates that at this value, the increase in revenue peaks and starts declining with further increases in advertising spend. It directly illustrates how the additional amount spent becomes less effective at increasing profits.

Rate of Change

The rate of change, which is computed by the first derivative, explains how quickly revenue is rising or falling as advertising expenditures adjust. It aids in intuitively understanding how efficient your spending is at any point.

If you observe the rate of change and notice its decrease over time, it signals the approaching point of diminishing returns. As determined from our previous analyses, before reaching \( x = 200 \), the revenue increases effectively with more spending.

However, beyond this point, the rate of change decreases despite additional spending, indicating that you get less bang for your buck, so to say. Thus, when planning advertising budgets, it's crucial to consider this rate to avoid excessive spending that minimally impacts revenue.