Identify the point of diminishing returns for the input-...

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.
$R=-\frac{4}{9}\left(x^{3}-9 x^{2}-27\right), \quad 0 \leq x \leq 5$

Key Concepts

Optimization Using Calculus

Optimization in this context uses calculus to find the input level at which the revenue function reaches a local maximum or where the rate of increase in revenue starts to slow down. The application of first and second derivatives helps pinpoint the precise point where the marginal increase in revenue declines, facilitating decision-making in resource allocation.

Marginal Analysis

Marginal analysis involves examining the derivative of the revenue function with respect to the input to assess the incremental change in revenue as the input increases. This analysis is essential for identifying the point where additional expenditure begins to yield less benefit, thus signaling diminishing returns.

Graphical Verification

Graphical verification involves using a graphing utility to plot the revenue function and visually inspect the location of the point of diminishing returns. This step confirms the analytical results by demonstrating the behavior of the function, making it easier to understand the relationship between input and output.

Revenue Function

This is a mathematical representation of revenue as a function of input variables (such as advertising expenditure). It models how revenue is generated based on the specific amount spent, and its form (often polynomial) provides insights into the relationship between investment and returns.

Diminishing Returns

The concept of diminishing returns refers to the point at which an increase in input (for example, advertising expenditure) yields progressively smaller increases in output (revenue). This is critical in determining the most efficient allocation of resources, as beyond this point the cost of additional input outweighs the benefits.

Transcript

Please give Ace some feedback