math question help

Question

The cost function for a telephone manufacturing plant is 𝐶(𝑥) = 5𝑥 + 24 and the revenue function is 𝑅(𝑥) = −2𝑥2 + 27𝑥, where 𝑥 is the number of telephones sold, in thousands. Determine the number of telephones sold that will maximize profit, 𝑃(𝑥) in thousands. State the maximum profit

Answer 1

P(x) = R(x)- C(x) = -2x^2 +27x -5x -24

= -2x^2 + 22x -24

P'(x) = -4x +22 = 0

x = 22/4 = 5.5 thousand = 5,500

max Profit= P(5.5) = -2(5.5)^2 +22(5.5) -24

= -60.5 + 121 -24

= 121 -84.5

= $36.5 thousand