Answer
$x=120$
Work Step by Step
The cost function is given as $C(x)=0.002x^3+9x+6912$.
The minimum average cost per unit is given by
$\bar{C(x)}=\frac{C(x)}{x}$
$\bar{C(x)}=0.002x^2+9+\frac{6912}{x}$
To find the critical points, we equate the first derivative to 0.
$\bar{C(x)}':=0.004x-\frac{6912}{x^2}=0\implies x=120$
To verify if $\bar{C(x)}$ is minimum at $x=120$ we check the derivatives on both sides of $x=120$.
$\bar{C(100)'}=0.4-0.6912<0$
$\bar{C(200)'}=0.8-0.1728>0$
The sign of derivative changes from $-$ to $+$ about $x=120$. Therefore, the average cost per unit is minimum at $x=120$.
