I think the answer is that although costs are minimum when the MC curve intersects the AC curve (i.e the last additional unit of output costs the same as the overall average unit of output) profit is not. More money can be made by producing less.
We follow the marginal revenue curve, which tends to fall as output increases, to understand how much cash the company is making. At the point where costs are lowest I.e MC=AC, marginal revenue could have fallen greatly fallen, since MC is now much greater than MR, despite average costs being at a minimum.
Another way of explaining it is through the firm's long-run output decision, which uses a two-stage argument:
I) Firstly, the firm needs to find the best profit-maximising output. They use MC=MR (marginal condition
)to find it I.e the cost of producing the last unit is equal to the revenue gained from producing that output. Because costs go up and revenue goes down, the next unit will cost more than than the company receives, so they stop here at this intersection.
Ii) With the MC=MR output decision made the firm must now decide through analysis whether this output is good enough to stay in business in the long run. They use the average condition
and compare the price/average revenue
they will receive at this MC=MR output with the average cost.
If the price they receive is above the average cost at this MC=MR output then they can carry on producing and stay in business.
If the price they receive is below the average cost at the MC=MR output then these losses will cause them to shut down.
So therefore MC=AC isn't the maximum profit that could be gained but the output at which minimum cost occurs. Knowledge of revenue and marginal revenue (MR) is needed to figure out how to bring in the greatest amount of cash.
Hope that helps; I've found most Economics textbooks to be generally poor at explaining Microeconomic stuff.
You really need a diagram too.
Average revenue and price are the same thing