If `C(x) = 3x^(2) - 2x + 3`, the marginal cost when x = 3 is

Options

A. 24
B. 16
C. 11
D. none of the above

Answer

To find the marginal cost when \( x = 3 \) for the cost function \( C(x) = 3x^2 - 2x + 3 \), we will follow these steps:

### Step 1: Identify the Cost Function
The cost function is given as:
\[ C(x) = 3x^2 - 2x + 3 \]

### Step 2: Find the Marginal Cost Function
The marginal cost function \( M(x) \) is the derivative of the cost function \( C(x) \) with respect to \( x \). We will differentiate \( C(x) \):
\[ M(x) = \frac{d}{dx} C(x) = \frac{d}{dx} (3x^2 - 2x + 3) \]

### Step 3: Differentiate the Cost Function
Using the power rule of differentiation:
\[ \frac{d}{dx} (3x^2) = 6x, \quad \frac{d}{dx} (-2x) = -2, \quad \frac{d}{dx} (3) = 0 \]
Thus, the marginal cost function becomes:
\[ M(x) = 6x - 2 \]

### Step 4: Evaluate the Marginal Cost at \( x = 3 \)
Now, we substitute \( x = 3 \) into the marginal cost function:
\[ M(3) = 6(3) - 2 \]
Calculating this gives:
\[ M(3) = 18 - 2 = 16 \]

### Conclusion
The marginal cost when \( x = 3 \) is:
\[ \boxed{16} \]