The demand function for a well known economics textbook is:

Question

P = 100 - .005Q
The publisher must pay $20 per book in printing and distribution costs and, in
addition, it must pay the author a $20 royalty for each book sold.
(a) Your job is to provide advice to the publisher. What price will maximise
the publisher’s profit? How much profit will the publisher earn? What will
be the total royalty payment earned by the author?
(b) A consultant says that the publisher and the author have the wrong type of
agreement. He says that the author and the publisher should tear up their
original agreement, in which the author gets $20 per book sold, and enter
into a profit-sharing agreement. He recommends that the author gets 40%
of the profit and the publisher 60%. What price should the publisher set
with this profit-sharing agreement?
(c) Will both the author and the publisher prefer the profit-sharing agreement
to their original agreement? Which agreement will the students who buy
the textbook prefer?
(d) Given the demand and cost conditions indicated above suppose that the
royalty payment was such that the author received a payment which was
15% of sales revenue. Prove that there is an inherent conflict between the
author and the publisher in that the author has an interest in the book’s
price being lower than the price which maximises the publisher’s profit.

Take a shot.

Note that the publisher is a monopolist wrt to the textbook. Always maximize where marginal cost=marginal revenue. So, for each sub-problem, construct a marginal cost and marginal revenue equations.