# What is max here, and how did they derive the first order condition from it?

Hello. Someone sent this on a discord server that I am on, and I couldn't make heads or tails of this. I found the book it is from on studocu; solutions to Cabral's Industrial Organisations, 2nd Ed. The question has MC = 0.

Is max p just max profit? Because then it would make some sense for me. And how did they derive the first order condition from this? All I found was that (∂π / ∂q) = 0, but please do explain where this comes from? Also, why is consumer surplus just the square of the total demand from a type 1 consumer?

Thank you so much in advance!

$$max_p f(p)$$ just means the maximal value of expression of $$f(p)$$ with regards to $$p$$ (meaning the value of $$f(p)$$ is checked over the whole range of $$p$$'s possible values).
The given first order condition is a derivative of the expression with respect to $$p$$, divided by 3. Division by 3 is fine since the right hand side of the first order condition is 0, which divided by 3 will remain 0.