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.
$R=\frac{1}{50,000}\left(600 x^{2}-x^{3}\right), \quad 0 \leq x \leq 400$

Key Concepts

Derivatives and Optimization

In calculus, finding the first derivative of a function helps identify critical points where the output is optimized. In this context, the first derivative represents the marginal revenue and its maximum indicates the optimal spending point before diminishing returns set in. The second derivative is used to analyze the concavity of the function, confirming whether the marginal revenue is decreasing.

Diminishing Returns

This concept refers to the point at which adding more input, such as spending on advertising, results in progressively smaller increases in output, such as revenue. It is a key principle in both economics and calculus, where the rate of improvement slows down after a certain level of input is reached.

Marginal Analysis

Marginal analysis involves evaluating the change in output produced by a small change in input. Mathematically, it is represented by the first derivative of the revenue function, which gives the marginal revenue. The point of diminishing returns is characterized by the maximum value of this marginal revenue before it starts to decline.

Graphical Analysis

Graphing utilities are used to visually inspect the behavior of the revenue function. By plotting the function and its derivative, one can confirm the point at which the rate of increase (slope) begins to decline, thus verifying the analytically determined point of diminishing returns.

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