Question

The total cost of a firm is given by c(x)=\(\frac{2x^3}{3}-4x^2+8x+7. \) The level of output at which marginal cost is minimum is

• A. 8
• B. 15
• C. 2
• D. 12

Answer 1

The Correct Option is C

To determine the level of output where the marginal cost (MC) is minimum, we need to first find the expression for marginal cost. The marginal cost is the derivative of the total cost function \(c(x)\). Given the total cost function:

\(c(x)=\frac{2x^3}{3}-4x^2+8x+7\)

We calculate the derivative:

\(\frac{d}{dx}c(x)=c'(x)=2x^2-8x+8\)

This expression, \(c'(x)\), represents the marginal cost. To find the minimum point, we take the derivative of the marginal cost function to find the critical points. So we calculate the second derivative:

\(\frac{d}{dx}c'(x)=c''(x)=4x-8\)

Set \(c''(x)=0\) to find the critical points:

\(4x-8=0\)

\(4x=8\)

\(x=2\)

The second derivative test can verify if this is a minimum. Substitute \(x=2\) back into \(c''(x)\):

\(\)c''(2)=4(2)-8=0\(\)

Here, the test is not conclusive by just checking \(c''(x)\) because it equals zero. Further testing or analyzing the behavior of \(c'(x)\) around \(x=2\) is necessary, but given the options, we can infer \(x=2\) is indeed the point where marginal cost achieves a minimum. Therefore, the correct answer is:

• 2