Minimizing average cost per unit

First express the average cost function by dividing each term of $C(q)$ by $q$: $\bar{C}(q) = \frac{500}{q} + 4 + 0.001q.$ This is now a function of the single variable $q$, with natural domain $q > 0$ since production levels must be positive. To optimize it, differentiate: $\bar{C}'(q) = -\frac{500}{q^2} + 0.001.$

Setting $\bar{C}'(q) = 0$ gives $\frac{500}{q^2} = 0.001$, so $q^2 = 500{,}000$, giving $q = \sqrt{500000} \approx 707.1$ (rejecting the negative root since $q>0$). Since $\bar{C}''(q) = 1000/q^3 > 0$ for $q>0$, this critical point is a minimum. Substituting back: $\bar{C}(707.1) = \frac{500}{707.1} + 4 + 0.001(707.1) \approx 0.707 + 4 + 0.707 \approx 5.41.$ So the minimum average cost per unit is about \5.41$, occurring at approximately$q \approx 707$ units.