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

Answer: (C)

Max sales = $(\frac{3}{4})^\frac{2}{3}$A

Answer 2

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 $\left(\frac{3}{4}\right)^{\frac{2}{3}}A$.