If the total cost function is given by `C(x) = 10x - 7x^(2) + 3x^(3)`, then the marginal average cost

Options

A. `10-14x+9x^(2)`
B. `10-7x+3x^(2)`
C. `-7+6x`
D. `-14+18x`

Answer

To find the marginal average cost from the given total cost function \( C(x) = 10x - 7x^2 + 3x^3 \), we will follow these steps:

### Step 1: Find the Average Cost Function
The average cost function \( AC(x) \) is given by the total cost function divided by the quantity \( x \):
\[ AC(x) = \frac{C(x)}{x} = \frac{10x - 7x^2 + 3x^3}{x} \]
Simplifying this, we get:
\[ AC(x) = 10 - 7x + 3x^2 \]

### Step 2: Differentiate the Average Cost Function
Next, we need to find the marginal average cost, which is the derivative of the average cost function with respect to \( x \):
\[ \text{Marginal Average Cost (MAC)} = \frac{d}{dx} AC(x) \]
Calculating the derivative:
\[ \frac{d}{dx}(10 - 7x + 3x^2) = 0 - 7 + 6x \]
Thus, we have:
\[ MAC = -7 + 6x \]

### Final Result
The marginal average cost is:
\[ MAC = -7 + 6x \]