The cost and price functions for a new Internet search company are reasonably approximated by the following models:
$C(x) = 37 + 1.42x – 0.0067x^2 + 0.00011x^3$
$p(x) = 3.7 – 0.007x$
Where $x$ represents the number of searches (in millions) per month. Find the ideal number of searches this company can manage (to maximize profit) per month.
(A) Approximately 65 million searches per month
(B) Approximately 76 million searches per month
(C) Approximately 82 million searches per month
(D) Approximately 97 million searches per month
(E) None of the above
So I believe the function to be maximized would be the price function minus the cost function:
Profit function, $G(x) = p(x)-C(x) = 3.7 - 0.007x - 37-1.42x+0.0067x^2 -0.00011x^3$
$G'(x) = 0 -0.007 -0-1.42 +0.0134 x -0.00033x^2 = 0$
I've been getting complex imaginary numbers here and I was wondering if I did something wrong or if the answer is actually none of the above. It seems like there should be real answers but I don't know how to explain why if there isn't.