Determine the production level that will result in the smallest average production cost.

Question

Minimizing Average Cost

 The total monthly cost (in dollars) incurred by a certain guitar manufacturer in manufacturing x units of its Professional Series guitars is given by the function

C(x) = 0.002x² + 100x + 8000.

(a)

Find the average cost function C.

C =  

0.002x+8000x​+100

 

 

Answer 1

Average cost C_av(x) = C(x)/x = 0.002x + 100 + 8000/x;

C_av(x)' = 0.002 - 8000/x² = 0; x² = 8000/0.002 = 4000000; x = 2000

To prove that this is minimum let find C_av(x)'' = 16000/x³; C_av'''(2000) = 0.000002 > 0.

By second derivative test we have that production level x = 2000 gives minimum average cost