The total cost function is given by `C(x) = 2x^(3)-3.5x^(2) +x`. Find the marginal average cost function.

Answer

To find the marginal average cost function from the given total cost function \( C(x) = 2x^3 - 3.5x^2 + x \), we will follow these steps:

### Step 1: Define the Average Cost Function
The average cost function \( A(x) \) is defined as the total cost function \( C(x) \) divided by the quantity \( x \):
\[ A(x) = \frac{C(x)}{x} \]
Substituting the given total cost function:
\[ A(x) = \frac{2x^3 - 3.5x^2 + x}{x} \]

### Step 2: Simplify the Average Cost Function
Now, we simplify the average cost function:
\[ A(x) = \frac{2x^3}{x} - \frac{3.5x^2}{x} + \frac{x}{x} \]
\[ A(x) = 2x^2 - 3.5x + 1 \]

### Step 3: Differentiate the Average Cost Function
To find the marginal average cost function, we need to differentiate the average cost function \( A(x) \) with respect to \( x \):
\[ \text{Marginal Average Cost} = \frac{d}{dx} A(x) = \frac{d}{dx}(2x^2 - 3.5x + 1) \]

### Step 4: Apply the Derivative Rules
Using the power rule of differentiation \( \frac{d}{dx}(x^n) = nx^{n-1} \):
\[ \frac{d}{dx}(2x^2) = 2 \cdot 2x^{2-1} = 4x \]
\[ \frac{d}{dx}(-3.5x) = -3.5 \cdot 1 = -3.5 \]
\[ \frac{d}{dx}(1) = 0 \]

### Step 5: Combine the Results
Now we combine the results of the differentiation:
\[ \text{Marginal Average Cost} = 4x - 3.5 \]

### Final Answer
Thus, the marginal average cost function is:
\[ \text{Marginal Average Cost} = 4x - 3.5 \]

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