Math Problem Statement

Find the point of diminishing returns (x, y) for the function R(x) = 11,000 - x^3 + 36x^2 + 700x, where x represents the amount spent on advertising (in thousands of dollars).

Solution

To solve the problem of finding the point of diminishing returns for the revenue function R(x)=11,000−x3+36x2+700xR(x) = 11,000 - x^3 + 36x^2 + 700x, we need to follow these steps:

1. Find the first derivative R′(x)R'(x):

The point of diminishing returns occurs when the rate of increase in revenue begins to slow down, meaning the second derivative becomes zero (inflection point). But first, we calculate the first derivative to find where the slope is zero (local maximum).

= -3x^2 + 72x + 700$$ ### 2. Find the second derivative $$R''(x)$$: We now find the second derivative to determine the point where the rate of revenue increase slows down: $$R''(x) = \frac{d}{dx}\left(-3x^2 + 72x + 700\right) = -6x + 72$$ ### 3. Set $$R''(x) = 0$$ to find the point of diminishing returns: Set the second derivative equal to zero and solve for $$x$$: $$-6x + 72 = 0$$ $$6x = 72$$ $$x = 12$$ ### 4. Find the corresponding revenue $$R(12)$$: Now substitute $$x = 12$$ into the original function $$R(x)$$ to find the revenue at this point: $$R(12) = 11,000 - (12)^3 + 36(12)^2 + 700(12)$$ $$R(12) = 11,000 - 1,728 + 36(144) + 8,400$$ $$R(12) = 11,000 - 1,728 + 5,184 + 8,400 = 22,856$$ Thus, the point of diminishing returns is $$(12, 22,856)$$, meaning when $12,000 is spent on advertising, the revenue reaches $22,856,000. Would you like more details or have any questions? ### Five related questions: 1. How do you find the local maximum or minimum of a function using the first derivative test? 2. What is the economic significance of the second derivative in this context? 3. Could diminishing returns occur earlier if the revenue function had a different cubic coefficient? 4. How does the concept of concavity relate to the point of diminishing returns? 5. What happens to revenue after the point of diminishing returns? ### Tip: In optimization problems, always check both the first and second derivatives to ensure you're identifying the correct type of critical point (maximum, minimum, or inflection).

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivatives
Optimization
Revenue Functions

Formulas

First derivative: R'(x) = -3x^2 + 72x + 700
Second derivative: R''(x) = -6x + 72

Theorems

Second Derivative Test: A zero value of the second derivative marks a point of inflection, indicating diminishing returns.

Suitable Grade Level

Grade 12 - College

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