Final answer:
The maximum profit for the function P(x) = 2000 - 6x^2 is found at the vertex of the parabola, which is x = 0. This suggests a charging price of $0, which is not reasonable and indicates an issue with the provided function or answer choices.
Step-by-step explanation:
To determine how much the company should charge per item (x) to have maximum profits for the function P(x) = 2000 - 6x2, we need to find the vertex of the parabola represented by the quadratic function. This is because the vertex of a parabola represented by a quadratic function ax2 + bx + c is the point of maximum or minimum value of the function, and since the coefficient of x2 is negative (a=-6), this parabola opens downward which means the vertex will give us the maximum profit.
To find the vertex, we can use the vertex formula x = -b/(2a). In this case, our equation has no b value and our a is -6, so x = 0/(2*(-6)) = 0. Therefore, the company should charge $0 per item, which does not make sense in a real-world scenario. There seems to be an error in the question, as no valid option corresponds to the correct answer based on the given profit function. The options provided do not match the vertex of the parabola, indicating a possible mistake in the provided function or the answer choices.