Given that the total cost junction for x units of a commodity is: `C(x) = x^(3)/3 + 3x^(2) - 7x + 16`. Find the Marginal Cost (MC) .

Answer

To find the Marginal Cost (MC) from the given total cost function \( C(x) = \frac{x^3}{3} + 3x^2 - 7x + 16 \), we need to follow these steps:

### Step 1: Understand the Marginal Cost
The Marginal Cost (MC) is defined as the derivative of the total cost function with respect to the number of units produced, \( x \). Mathematically, this can be expressed as:
\[ MC(x) = \frac{dC}{dx} \]

### Step 2: Differentiate the Total Cost Function
We will differentiate the given total cost function \( C(x) \):
\[ C(x) = \frac{x^3}{3} + 3x^2 - 7x + 16 \]

Using the power rule of differentiation, which states that \( \frac{d}{dx}[x^n] = nx^{n-1} \), we differentiate each term:

1. **Differentiate \( \frac{x^3}{3} \)**:
\[ \frac{d}{dx}\left(\frac{x^3}{3}\right) = \frac{3x^2}{3} = x^2 \]

2. **Differentiate \( 3x^2 \)**:
\[ \frac{d}{dx}(3x^2) = 6x \]

3. **Differentiate \( -7x \)**:
\[ \frac{d}{dx}(-7x) = -7 \]

4. **Differentiate the constant \( 16 \)**:
\[ \frac{d}{dx}(16) = 0 \]

### Step 3: Combine the Derivatives
Now, we combine all the derivatives we calculated:
\[ MC(x) = x^2 + 6x - 7 \]

### Final Answer
Thus, the Marginal Cost function is:
\[ MC(x) = x^2 + 6x - 7 \]

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