Questions: For the cost function C(x)=25600+400x+x^2. Use the formula C̄(x)=C(x)/x to compute the average cost. Then, find the zero of its derivative to find the production level x minimizing the average cost. The production level that minimizes the average cost is

For the cost function C(x)=25600+400x+x^2. Use the formula C̄(x)=C(x)/x to compute the average cost. Then, find the zero of its derivative to find the production level x minimizing the average cost.

The production level that minimizes the average cost is

Transcript text: For the cost function $C(x)=25600+400 x+x^{2}$. Use the formula $\bar{C}(x)=\frac{C(x)}{x}$ to compute the average cost. Then, find the zero of its derivative to find the production level $\times$ minimizing the average cost. The production level that minimizes the average cost is $\square$

Solution

Solution Steps

Step 1: Define the Average Cost Function

The average cost function is $AC(x) = \frac{x^{2} + 400 x + 25600}{x} = \frac{x^{2} + 400 x + 25600}{x}$.

Step 2: Find the Critical Points

The derivative of $AC(x)$ with respect to $x$ is $\frac{d}{dx}AC(x) = \frac{2 x + 400}{x} - \frac{x^{2} + 400 x + 25600}{x^{2}}$. Setting this equal to zero gives the critical points.

Step 3: Determine the Minimum Average Cost

Since we are dealing with a quadratic function, the critical point at $x = 160$ is a minimum.