Am I doing something wrong here? Maximization of profit problem. 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, $G(x) = p(x)-C(x) = 3.7 - 0.007x - 37-1.42x+0.0067x^2 -0.00011x^3$
Maximize function:
$G'(x) = 0 -0.007 -0-1.42 +0.0134 x -0.00033x^2 = 0$
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.
 A: As I mentioned in my comment, your equations seem to be right, but you don't have the correct function for optimization - with my method below, Wolfram Alpha gives an acceptable answer.
You want to maximize profit, and you have the price function and the cost function. To maximize profit, you optimize by subtracting cost from revenue, and optimizing the resulting function with appropriate restrictions (i.e. $x>0$). Notice that:


*

*The price function gives the price for a single unit. Thus, to get revenue, we need to multiply the price function by $x$, so we need to use $x(3.7-0.007x)$ in the final equation, instead of $3.7-0.007x$.

*Why can't we use the price function? Because the price function here is simply giving us the price of a single unit as a function of the number of units that are sold.

*The cost function, however, is already implicitly (not in the mathematical sense) a function for total cost, since it takes into account the number of units we are selling. Thus, we don't need to modify it.
This means that we need to optimize $(3.7-0.007x)x-37-1.42x+0.0067x^2-0.000111x^3$, subject to appropriate constraints on $x$. You should get a reasonable result from this.
