I have a formula for the long-run total cost curve,
$$TC(Q) = 6000Q + 40Q^2 + Q^3$$
and I'm trying to find the quantity that minimizes the long-run average total cost.
I assume I'm trying to find the value $Q$ (quantity) that produces the total cost ($TC(Q)$) where $\frac{TC(Q)}{Q}$ is the lowest. But how do I do this short of trial and error?