Afirm produces a commodity and receives $100 for each unit sold. the cost of producing and selling x units is 20x + 0.25x 2 dollars. (a) find the production level which maximizes profit. (in other words, find the x that maximizes profit) (b) a tax of $10 per unit is imposed on the firm. what is the new optimum production level?

So for each unit they get $100, so if they sell "x" units, they intake 100x, that's just the gross income, and therefore the Revenue.

Now, they have a cost of 20x + 0.25x² if they "x" units, now, bearing in mind that if you pluck out the costs out of the incoming revenue, what's leftover, is a surplus amount, namely the Profit.  Therefore, profit p(x) = revenue - costs, or  100x - (20x + 0.25x²), which is p(x) = -0.25x² +80x.

a)
now, what's the highest profit they can make?  check the picture below.



b)

well, if a tax is imposed on them, and say, they take it out of the revenue, then their new revenue is not 100x, is 90x, thus   

and you can use the same formula as above to get the x-coordinate for that vertex.