A question on revenue function for calculus

Question

A certain band charges $30 per ticket for concerts when 297 people attend. The band members realize that for every $10 increase in the ticket price 11 fewer people will attend the concert. Find a revenue function R in terms of x where x represents the number of additional $10 increases in the ticket price. Find R(x), find R'(x), and find R''(x).

Answer 1

R(x) = (297 - 11x) (30 + 10x) = 8910 + 2640x - 110x²;

R'(x) = 2640 - 220x. To get maximum revenu we set R'(x) = 2640 - 220x = 0, so x = 2640/220 = 12

So max revenu R_max = R(12) = 24750 if price = 30 + 120 = $150 per ticket

R''(x) = - 220 < 0 so we have max by second derivative test.