Question

The cost function at production x is defined as C (x) = 3x³- x + 2 and sale function at A cost x is defined as S(x) = \((\frac{A}{x^\frac{1}{3}})\). Which of the following is true?

• A. Min sales = \((\frac{3}{4})^\frac{2}{3}\)A
• B. Min sales = \((\frac{9}{2})^\frac{2}{3}\)A
• C. Max sales = \((\frac{3}{4})^\frac{2}{3}\)A
• D. Max sales = \((\frac{9}{2})^\frac{2}{3}\)A

Answer 1

The Correct Option is C

The correct option is (C): Max sales = \((\frac{3}{4})^\frac{2}{3}\)A
Explanation: Given the cost function \(C(x) = 3x^3 - x + 2\) and the sales function \(S(x) = \frac{A}{x^{\frac{1}{3}}}\), we are tasked with determining which of the options regarding minimum or maximum sales is true.
The correct answer is **C: Max sales = \((\frac{3}{4})^{\frac{2}{3}}\)A**.
To understand this:
- The cost function is a cubic function, and the sales function \(S(x) = \frac{A}{x^{\frac{1}{3}}}\) indicates the relationship between sales and production, where \(x\) is the level of production.
- By examining the sales function, you can identify that sales decrease as production increases (since \(S(x)\) is inversely proportional to \(x^{1/3}\)).
- The maximum sales occur when production is at an optimal lower value, leading to the form given by \(\left(\frac{3}{4}\right)^{\frac{2}{3}}A\).