Answer
See explanations.
Work Step by Step
Step 1. Assume the cost of producing $x$ items is $c(x$); the average cost is then $A_c=\frac{c(x)}{x}$. Marginal cost is defined as $M_c=c'(x)$
Step 2. To minimize $A_c$, take its derivative and get $A'_c=\frac{xc'(x)-c(x)}{x^2}$. Let it be zero to get $xc'(c)=c(x), x=\frac{c(x)}{c'(x)}$ and thus $A_c=\frac{c(x)}{x}=c'(x)$
Step 3. Check the signs of $A'_c$ across the critical point to get $..(-)..(\frac{c(x)}{c'(x)})..(+)..$ and we know it is a minimum.
