A little Cost Problem

Economics have alot concepts that uses calculus. Finding Breakeven Price for the company is one of them. Break-even means profit earned = 0. Why is this the case? Lets look at the Cost function: Total Cost = Total Fixed Cost + Total Variable Cost. Fixed Cost means that it is incurred before production occurred. Variable Cost means the cost that is incurred as production increased.

How it will link to profit you will say? Profit = Total Revenue - Total Cost. Total Revenue = Price of the Product * Quantity Sold Total Cost = Average Cost per Unit * Quantity Sold. When Profit = 0 though: Total Revenue = Total Cost. Price of the Product* Q = Average Cost per Unit * Q Therefore: Price = Average Cost. ((For those who dont know how to derive the average cost, it means TC / q, because you are finding on average how much cost is incurred per unit.))

Surprising though, This point is also the minimum point for the AC function. Why you will say? Lets think about the logic: Fixed cost incurred before production starts, and average of the fixed costs decreases as production increases. Variable cost increases as more is produced. So, here is the real question: Given that: TC(q) = $q \times (ln(q)^{2}) - 2q^{2}$ , find the minimum point on the average cost function. (Minimum Quantity)