Profit maximization, when $MC = P$, in a simple supply table and curve Our economics school book states that the profits of a company are maximized when MC = P.
However, I am having some trouble wrapping my head around that.
For example, I have the following exercise:



The question is: If the equilibrium p is $P_E = 72$, (in which the product is sold), what quantity $Q$ will the company have to produce and sell in order to maximize its profit?   

So, we get that the supply table is:



The answer is: it will have to produce and sell quantity of $Q_E = 450$.
My questions are:


Is this because of the equilibrium p, which is $P_E=72$ equals the $MC = 72$?
If the equilibrium price was set as $P_E = 36$, then would the answer be $Q_E=400$?
If the equilibrium price was, for example, $P_E=71$, then would I be able to answer just with this data?


Thanks for your time!!
What your school economics textbook must say is that : 

"In a Perfectly Competitive Market (PC), firms maximize profits when MC= P ". This equality applies only to firms in a perfectly competitive market. 

This is because, in a Perfectly Competitive Market there is an infinite number of buyers and sellers and the prices are set by the overall market forces, and the firms and the consumers take that price to be given. 

Using Calculus, it can easily be shown that MC=P is the condition for Profit Maximization in PC markets. 


We can further see other conditions of maxima using second-order conditions (which relate the slope of the MC curve in relation to the slope of MR curve ) but we can ignore those for now. 

Intuitively, a producer earns P on each unit sold. If the MC &lt; P, they earn more on each marginal unit than they spend on its production, and hence they would keep producing more. If MC > P, then the cost of producing each marginal unit is more than the revenue it fetches and hence they would cut down on production. Neither of these is a position of equilibrium for the firm. Hence MC=P is the position of equilibrium and the First Order Condition for profit maximization.


Yes
Yes
Yes, you can't answer this question, because the data set given to you is made up of discrete points.