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

Answer

To find the range of output for which the average cost function \( AC \) is decreasing, 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: Differentiate the average cost function
To find where the function is decreasing, we need to differentiate \( AC \) with respect to \( x \):
\[ \frac{d(AC)}{dx} = \frac{d}{dx}(2x - 11 + \frac{50}{x}) \]
Using the power rule and the quotient rule, we differentiate each term:
- The derivative of \( 2x \) is \( 2 \).
- The derivative of \( -11 \) is \( 0 \).
- The derivative of \( \frac{50}{x} \) can be rewritten as \( 50x^{-1} \), which gives \( -\frac{50}{x^2} \).

Thus, the derivative is:
\[ \frac{d(AC)}{dx} = 2 - \frac{50}{x^2} \]

### Step 3: Set the derivative equal to zero to find critical points
To find the critical points, we set the derivative equal to zero:
\[ 2 - \frac{50}{x^2} = 0 \]
Rearranging gives:
\[ \frac{50}{x^2} = 2 \]
Multiplying both sides by \( x^2 \) results in:
\[ 50 = 2x^2 \]
Dividing by \( 2 \):
\[ x^2 = 25 \]
Taking the square root gives:
\[ x = 5 \quad \text{or} \quad x = -5 \]

### Step 4: Determine intervals for increasing and decreasing
Now we will test the intervals determined by the critical points \( x = -5 \) and \( x = 5 \):
1. **Interval \( (-\infty, -5) \)**: Choose \( x = -6 \):
\[ \frac{d(AC)}{dx} = 2 - \frac{50}{(-6)^2} = 2 - \frac{50}{36} = 2 - \frac{25}{18} = \frac{36}{18} - \frac{25}{18} = \frac{11}{18} > 0 \]
(Increasing)

2. **Interval \( (-5, 5) \)**: Choose \( x = 0 \) (note that \( x \) cannot be zero, so we can choose a value between -5 and 5, e.g., \( x = -1 \)):
\[ \frac{d(AC)}{dx} = 2 - \frac{50}{(-1)^2} = 2 - 50 = -48 < 0 \]
(Decreasing)

3. **Interval \( (5, \infty) \)**: Choose \( x = 6 \):
\[ \frac{d(AC)}{dx} = 2 - \frac{50}{(6)^2} = 2 - \frac{50}{36} = 2 - \frac{25}{18} = \frac{36}{18} - \frac{25}{18} = \frac{11}{18} > 0 \]
(Increasing)

### Step 5: Conclusion
The average cost function \( AC \) is decreasing in the interval:
\[ (-5, 5) \]