Chapter 2.4, Problem 60E

Minimum average cost. Financial analysts in a company that manufactures DVD players arrived at the following daily cost equation for manufacturing $x$ DVD players per day:

$C\left(x\right)=x^2+2x+2,000$

The average cost per unit at a production level of $x$ players per day is $\bar{C}\left(x\right)=C\left(x\right)/x$.

(A) Find the rational function $\bar{C}$.

(B) Sketch a graph of $\bar{C}\text{ for }5\le x\le 150$.

(C) For what daily production level (to the nearest integer) is the average cost per unit at a minimum, and what is the minimum average cost per player (to the nearest cent)? [Hint: Refer to the sketch in part (B) and evaluate $\bar{C}\left(x\right)$ at appropriate integer values until a minimum value is found.]

(D) Graph the average cost function $C$ on a graphing calculator and use an appropriate command to find the daily production level (to the nearest integer) at which the average cost per player is at a minimum. What is the minimum average cost to the nearest cent?