The average cost function associated with producing and marketing x units of an item is given by `AC = 3x - 11+ (10)/(x)`. Then MC at x = 2 is

Options

A. 2
B. 5
C. 3
D. 1

Answer

To find the marginal cost (MC) at \( x = 2 \) given the average cost function \( AC = 3x - 11 + \frac{10}{x} \), we will follow these steps:

### Step 1: Write down the average cost function
The average cost function is given as:
\[ AC = 3x - 11 + \frac{10}{x} \]

### Step 2: Calculate the total cost function
The total cost (TC) can be found by multiplying the average cost (AC) by the number of units produced (x):
\[ TC = AC \times x = \left(3x - 11 + \frac{10}{x}\right) \times x \]
Distributing \( x \) gives:
\[ TC = 3x^2 - 11x + 10 \]

### Step 3: Find the marginal cost function
Marginal cost (MC) is the derivative of the total cost function with respect to \( x \):
\[ MC = \frac{d(TC)}{dx} = \frac{d(3x^2 - 11x + 10)}{dx} \]
Calculating the derivative:
\[ MC = 6x - 11 \]

### Step 4: Evaluate the marginal cost at \( x = 2 \)
Now, we substitute \( x = 2 \) into the marginal cost function:
\[ MC = 6(2) - 11 = 12 - 11 = 1 \]

### Final Answer
Thus, the marginal cost at \( x = 2 \) is:
\[ \boxed{1} \]