For the cost function , show that average costs are minimized at the x-value where by finding this minimum. Round your answer to two decimal places. (Note: )

Question

For the cost function  , show that average costs are minimized at the x-value where  by finding this minimum. Round your answer to two decimal places. (​Note:  ) ​​

Quizplus · Accepted Answer

To find the minimum of the average cost function, we need to first find the derivative of the cost function with respect to x and set it equal to 0:

f'(x) = 500 - 0.1x - 0.004x^2 = 0

Solving for x, we get x = 6250.

To confirm that this point gives us a minimum, we can take the second derivative:

f''(x) = -0.008x

At x = 6250, f''(x) < 0, which means that the function is concave down and has a minimum at that point.

Finally, to find the corresponding average cost, we plug x = 6250 back into the original cost function and divide by x:

AC(x) = (40000 + 500x + 0.02x^2) / x

AC(6250) = $8.12

Therefore, the correct choice is C.

For the Cost Function , Show That Average Costs