Chapter 9.4, Problem 53E

Cost and average cost Suppose that the total cost function, in dollars, for the production of x units of a product is given by

$C(x)=4000+55x+0.1x^2$

Then the average cost of producing x items is

$\overline{C(x)}=\frac{\text{total cost}}{x}=\frac{4000}{x}+55+0.1x$

(a) Find the instantaneous rate of change of average cost with respect to the number of units produced at any level of production.

(b) Find the level of production at which this rate of change equals zero.

(c) At the value found in part (b), find the instantaneous rate of change of cost and find the average cost. What do you notice?