Given the total cost function for x units of commodity as `C(x)=1/3 x^3+3x^2-16x+2` . Find Marginal cost function .

Answer

To find the marginal cost function from the given total cost function \( C(x) = \frac{1}{3} x^3 + 3x^2 - 16x + 2 \), we need to differentiate the total cost function with respect to \( x \).

### Step-by-Step Solution:

1. **Write down the total cost function:**
\[ C(x) = \frac{1}{3} x^3 + 3x^2 - 16x + 2 \]

2. **Differentiate the total cost function to find the marginal cost function:**
The marginal cost function \( MC(x) \) is given by:
\[ MC(x) = \frac{dC}{dx} \]

3. **Differentiate each term of \( C(x) \):**
- The derivative of \( \frac{1}{3} x^3 \) is:
\[ \frac{d}{dx} \left( \frac{1}{3} x^3 \right) = \frac{1}{3} \cdot 3x^2 = x^2 \]
- The derivative of \( 3x^2 \) is:
\[ \frac{d}{dx} (3x^2) = 3 \cdot 2x = 6x \]
- The derivative of \( -16x \) is:
\[ \frac{d}{dx} (-16x) = -16 \]
- The derivative of the constant \( 2 \) is:
\[ \frac{d}{dx} (2) = 0 \]

4. **Combine the derivatives:**
Now, we combine all the derivatives:
\[ MC(x) = x^2 + 6x - 16 \]

5. **Final result:**
Therefore, the marginal cost function is:
\[ MC(x) = x^2 + 6x - 16 \]