# Elementary Optimization Problem

I'm trying to solve this elementary calculus problem. It reads:

"You have written a compelling memoir called At My Limit: Learning Calculus During a Pandemic. According to your literary agent John, if you sell it at a price of 20 dollars per copy, 10,000 people will buy the book (resulting in a total income of 200,000 dollars). For every dollar you subtract from the book price, 4,000 additional people will buy the book. This also applies to fractional numbers of dollars. For example, if the book costs $19.90, 10,400 people will buy it. 1. If you are required to choose a book price of at least 10 dollars and no more than 15 dollars, determine the price that would result in the highest possible total income. Make sure to justify your answer using calculus, rather than intuition or techniques from other courses 2. If you are required to choose a book price of at least 5 dollars and no more than 10 dollars, determine the price that would result in the highest possible total income." I've deduced that for part 1, if the price per book is 15 dollars, there will be 30000 buyers, and thus a total income of$450000.

If the price per book is 10 dollars, there will be 50000 buyers, and the total income will be \$500000.

I don't know how I should go about setting up the equation and constraint, however.

Your income is $$I = (20-x)(10000+4000x) = 200000+70000 x-4000x^2$$ $$I' = 70000-8000x \overset{!}{=} 0 \quad \to x = 8.75$$ This price results in the maximum income and for the first interval it's the limit closer to this.