Question

A certain band charges $124 per ticket for concerts when 224 people attend. The band members realize that for every $6 increase in the ticket price 3 fewer people will attend the concert. Find a revenue function R in terms of x where x represents the number of additional $6 increases in the ticket price. [A] R(x) = Preview [B] Find R'(x) = Preview [C] Find the critical value(s): Preview [D] Perform the second derivative test to determine if there is an absoulte minimum at the critical value found. R''(2) Preview What is the best conclusion regarding an absolute maximum or minimum at this critical value. The second derivative test is inclusive with regards to an absolute maximum or minimum and the first derivative test should be performed Since R''(x) < 0 for all values of x on ( - 0, 0) we can colude that the R is concave down for all values of x on ( - 00, 0 )and that we therefore have an absolute maximum at the critical value on the entire domain (- 0,00 O At the critical value R''(x) < 0 so I can conclude that there is a local/relative minimum there but I can't conculde anything about an absolute minimum O At the critical value R''(x) < 0 so I can conclude that there is a local/relative maximum there but I can't conculde anything about an absolute maximum Since R''(x) < 0 for all values of x on ( -00,00) we can colude that the R is concave down for all values of x on ( - 00,00) and that we therefore have an absolute minimum at the critical value on the entire domain (- 0,00) [E] What should the ticket price be for the band to maximize its revenue? Preview [F] How many people will attend the concert when the revenue is maximized? Preview [G] What is the band's maximum revenue? Preview

Answer 1

al R(a) 224 – 3x)(124 + 6x) 18x² +972x + 27776 R'(a) = - 36 x + 972 Critical values when R'(x) = 0 :-36% +972 x = D Ril(a) = =36 last option Since R''() < 0 for all values of ron ( -00,00) we can colude that the R is concave down for all values of 2 on ( - 20,00) and that we therefore have an absolute minimum at the critical value on the entire domain (-00,00) E Ticket price 124 + 6(27) 286 to no. of people = 224 - 3x27 143 R max (143) *286 740898

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