A financial analyst is modeling a company's monthly profit, P, based on the number of units, x, sold. The profit function is given by P(x) = -2x^2 + 80x - 300. What is the maximum profit the company can achieve?

Correct answer: A. $900

The profit function is a quadratic equation P(x) = -2x^2 + 80x - 300, which is a parabola opening downwards (due to the negative coefficient of x^2). The maximum profit occurs at the vertex. The x-coordinate of the vertex is given by x = -b / (2a). For this function, a = -2 and b = 80, so x = -80 / (2 * -2) = -80 / -4 = 20. Now, substitute x = 20 back into the profit function to find the maximum profit: P(20) = -2(20)^2 + 80(20) - 300 = -2(400) + 1600 - 300 = -800 + 1600 - 300 = 800 - 300 = 500. Oh, wait, P(20) = -2(400) + 80(20) - 300 = -800 + 1600 - 300 = 800 - 300 = 500. This is a common error. Let me re-calculate: -800 + 1600 = 800. 800 - 300 = 500. My provided answer 'D' is incorrect based on my calculation. Let me correct the answer and explanation. The calculation is correct, so the answer should be $500. Let's re-verify. x = -b/(2a) = -80/(2*-2) = -80/-4 = 20. P(20) = -2(20)^2 + 80(20) - 300 = -2(400) + 1600 - 300 = -800 + 1600 - 300 = 800 - 300 = 500. The maximum profit is $500. The answer should be B. I will adjust the answer to B and explanation accordingly. Corrected Explanation: The profit function is a quadratic equation P(x) = -2x^2 + 80x - 300. Since the coefficient of x^2 is negative, the parabola opens downwards, meaning its vertex represents the maximum point. The x-coordinate of the vertex is found using the formula x = -b / (2a). Here, a = -2 and b = 80, so x = -80 / (2 * -2) = -80 / -4 = 20. To find the maximum profit, substitute x = 20 back into the profit function: P(20) = -2(20)^2 + 80(20) - 300 = -2(400) + 1600 - 300 = -800 + 1600 - 300 = 800 - 300 = 500. Thus, the maximum profit is $500.