Final answer:
The maximum income for the company is found by locating the vertex of the parabola described by the quadratic income function. The maximum income occurs at the price that corresponds to the vertex, resulting in a maximum income of $6,250,000.
Step-by-step explanation:
The function f(x) = -4x^2 + 10,000x is a quadratic function that represents the company's income from selling a product at a price x. To find the maximum income the company could make, we need to determine the vertex of the parabola. Since the coefficient of the x^2 term is negative, the parabola opens downwards, and the vertex represents the maximum point.
The x-coordinate of the vertex can be found using the formula -b/(2a), where a is the coefficient of x^2 and b is the coefficient of x. In our case, a = -4 and b = 10,000, so x = -10000/(2*(-4)) = 1250. Substituting x = 1250 back into the original function to find the maximum income yields f(1250) = -4(1250)^2 + 10000(1250) = -6250000 + 12500000 = 6250000.
Therefore, the maximum income the company could make selling the product is $6,250,000, which corresponds to option D.