A firm has the following total cost and demand functions :
`C(x) = (x^(3))/(3) - 10x^(2) + 300 x`, where x stand for output. Calculate output at which the
Average cost is minimum

Answer

To find the output at which the average cost is minimum, we will follow these steps:

### Step 1: Write the total cost function
The total cost function is given as:
\[ C(x) = \frac{x^3}{3} - 10x^2 + 300x \]

### Step 2: Write the average cost function
The average cost \( AC(x) \) is defined as the total cost divided by the quantity \( x \):
\[ AC(x) = \frac{C(x)}{x} = \frac{\frac{x^3}{3} - 10x^2 + 300x}{x} \]

### Step 3: Simplify the average cost function
Now, we simplify the average cost function by dividing each term by \( x \):
\[ AC(x) = \frac{x^3}{3x} - \frac{10x^2}{x} + \frac{300x}{x} = \frac{x^2}{3} - 10x + 300 \]

### Step 4: Differentiate the average cost function
To find the minimum average cost, we need to differentiate \( AC(x) \) with respect to \( x \) and set the derivative equal to zero:
\[ \frac{d(AC)}{dx} = \frac{d}{dx}\left(\frac{x^2}{3} - 10x + 300\right) \]
Calculating the derivative:
\[ \frac{d(AC)}{dx} = \frac{2x}{3} - 10 \]

### Step 5: Set the derivative equal to zero
Now, we set the derivative equal to zero to find the critical points:
\[ \frac{2x}{3} - 10 = 0 \]

### Step 6: Solve for \( x \)
Rearranging the equation gives:
\[ \frac{2x}{3} = 10 \]
Multiplying both sides by 3:
\[ 2x = 30 \]
Dividing by 2:
\[ x = 15 \]

### Conclusion
The output at which the average cost is minimum is:
\[ \boxed{15} \]