Calculus Help

Question

The total cost of production, in thousands of dollars, is given by C(q) = q3 – 12 q2 + 36q. where q is in thousands and 0 < q <8

a. Write the average cost function.

b. For what value of q is the average cost minimized?

Answer 1

The average cost function is the straight line connecting the two endpoints of the interval 0<q<8, which are (0,0) and (8,32).  The slope of the line is (32-0)/(8-0) = 4.  The y-intercept is 0 since one of the points on the line is (0,0).  So the equation of the average cost function is C_average(q) = 4q.

 

To find the minimum cost, take the derivative of C(q) wrt q, set it to zero, and solve for q:

 

C(q) = q³ - 12q² + 36q

C'(q) = 3q² - 24q + 36

0 = 3q^₂ - 24q + 36

0 = q² - 8q + 12      (divided both sides by 3)

0 = (q-2)(q-6)

 

So there are two extreme points, one at q = 2 and one at q = 6.  One point will be the maximum cost, the other the minimum cost.  To find which is which, you can either plug q=2 and q=6 into C(q) and see which is smaller; the q value that gives th4e smaller C(q) corresponds to the minimum cost. Or you can take the second derivative of C(q) then plug q = 2 and q = 6 into the second derivative.  The value that gives you a positive value for the second derivative corresponds to the minimum cost.