Given the total cost function for x units of a commodity as `C(x) = (1)/(3) x^(3) + 2x^(2) - 5x + 10`.

Answer

To solve the problem, we need to find the marginal cost function and the average cost function from the given total cost function \( C(x) = \frac{1}{3} x^3 + 2x^2 - 5x + 10 \).

### Step 1: Find the Marginal Cost Function

The marginal cost function is defined as the derivative of the total cost function with respect to \( x \).

1. **Differentiate the total cost function \( C(x) \)**:
\[ C'(x) = \frac{d}{dx} \left( \frac{1}{3} x^3 + 2x^2 - 5x + 10 \right) \]

2. **Apply the power rule for differentiation**:
- The derivative of \( \frac{1}{3} x^3 \) is \( x^2 \).
- The derivative of \( 2x^2 \) is \( 4x \).
- The derivative of \( -5x \) is \( -5 \).
- The derivative of a constant (10) is 0.

3. **Combine the derivatives**:
\[ C'(x) = x^2 + 4x - 5 \]

Thus, the marginal cost function is:
\[ \text{Marginal Cost} = C'(x) = x^2 + 4x - 5 \]

### Step 2: Find the Average Cost Function

The average cost function is defined as the total cost function divided by the number of units \( x \).

1. **Write the average cost function**:
\[ \text{Average Cost} = \frac{C(x)}{x} = \frac{\frac{1}{3} x^3 + 2x^2 - 5x + 10}{x} \]

2. **Simplify the expression**:
- Divide each term in the numerator by \( x \):
\[ \text{Average Cost} = \frac{1}{3} x^2 + 2x - 5 + \frac{10}{x} \]

Thus, the average cost function is:
\[ \text{Average Cost} = \frac{1}{3} x^2 + 2x - 5 + \frac{10}{x} \]

### Summary of Results
- **Marginal Cost Function**: \( C'(x) = x^2 + 4x - 5 \)
- **Average Cost Function**: \( \text{Average Cost} = \frac{1}{3} x^2 + 2x - 5 + \frac{10}{x} \)