Final answer:
To determine the maximum profit and corresponding output, calculate the profit function P(x) by subtracting the cost function C(x) from the revenue function R(x), then find the critical points of P(x), and finally analyze those points to find the maximum profit.
Step-by-step explanation:
To find the maximum profit and the number of units that must be produced and sold to yield the maximum profit with the given revenue R(x) = 7x - 2x^2 and cost C(x) = x^3 - 3x^2 + 2x + 1, we must first compute the profit function, which is profit, P(x) = R(x) - C(x).
Calculating the profit function:
P(x) = (7x - 2x^2) - (x^3 - 3x^2 + 2x + 1)
P(x) = -x^3 + 5x^2 - x - 1
To find the maximum profit, we can set the derivative P'(x) equal to zero to find the critical points, and then use the second derivative test or analyze the changes in sign of P'(x) to determine which critical point gives the maximum profit. The problem may require calculus tools such as differentiation to find the extremum point. However, given the options provided (4, 8, 12, 16), one would need to compute the profit for the relevant output levels to identify which one corresponds to the maximum profit.