The cost function at American Gadget is ` C(x)=x^(3)-6x^(2)+15x` (`x` is in thousands of units and ` x gt 0`) . The production level at which average cost is minimum is:

Options

A. (a) `2`
B. (b) `3`
C. (c) `5`
D. (d) `4`

Answer

To find the production level at which the average cost is minimum for the cost function \( C(x) = x^3 - 6x^2 + 15x \), we will follow these steps:

### Step 1: Define the Average Cost Function
The average cost \( A(x) \) is given by the total cost \( C(x) \) divided by the number of units produced \( x \):
\[ A(x) = \frac{C(x)}{x} = \frac{x^3 - 6x^2 + 15x}{x} = x^2 - 6x + 15 \]

### Step 2: Find the First Derivative
To find the minimum average cost, we first need to find the first derivative of \( A(x) \):
\[ A'(x) = \frac{d}{dx}(x^2 - 6x + 15) = 2x - 6 \]

### Step 3: Set the First Derivative to Zero
Next, we set the first derivative equal to zero to find the critical points:
\[ 2x - 6 = 0 \]
Solving for \( x \):
\[ 2x = 6 \quad \Rightarrow \quad x = 3 \]

### Step 4: Find the Second Derivative
To determine whether this critical point is a minimum or maximum, we find the second derivative:
\[ A''(x) = \frac{d^2}{dx^2}(x^2 - 6x + 15) = 2 \]

### Step 5: Analyze the Second Derivative
Since \( A''(x) = 2 \) is positive, this indicates that the function \( A(x) \) has a local minimum at \( x = 3 \).

### Conclusion
Thus, the production level at which the average cost is minimum is:
\[ \boxed{3} \]