The average cost function associated with producing and marketing x units of an items is given by `AC=2x-11+(50)/(x)`. Find the range of value of the output x, for which AC is increasing.

Answer

To find the range of values of the output \( x \) for which the average cost function \( AC \) is increasing, we will follow these steps:

### Step 1: Write down the average cost function
The average cost function is given by:
\[ AC = 2x - 11 + \frac{50}{x} \]

### Step 2: Find the derivative of the average cost function
To determine when the function is increasing, we need to find the derivative of \( AC \) with respect to \( x \):
\[ \frac{d(AC)}{dx} = \frac{d}{dx}(2x - 11) + \frac{d}{dx}\left(\frac{50}{x}\right) \]
Calculating the derivatives:
- The derivative of \( 2x - 11 \) is \( 2 \).
- The derivative of \( \frac{50}{x} \) can be calculated using the power rule:
\[ \frac{d}{dx}\left(\frac{50}{x}\right) = 50 \cdot \frac{d}{dx}(x^{-1}) = 50 \cdot (-1)x^{-2} = -\frac{50}{x^2} \]
Thus, we have:
\[ \frac{d(AC)}{dx} = 2 - \frac{50}{x^2} \]

### Step 3: Set the derivative greater than zero
To find when \( AC \) is increasing, we set the derivative greater than zero:
\[ 2 - \frac{50}{x^2} > 0 \]

### Step 4: Solve the inequality
Rearranging the inequality gives:
\[ 2 > \frac{50}{x^2} \]
Multiplying both sides by \( x^2 \) (assuming \( x > 0 \)):
\[ 2x^2 > 50 \]
Dividing both sides by 2:
\[ x^2 > 25 \]

### Step 5: Take the square root
Taking the square root of both sides, we get:
\[ |x| > 5 \]
This implies two ranges:
\[ x < -5 \quad \text{or} \quad x > 5 \]

### Conclusion
Since \( x \) represents the number of units produced, we typically consider only the positive range in this context. Therefore, the average cost function \( AC \) is increasing for:
\[ x > 5 \]