Final answer:
To find the maximum profit and the number of items that must be sold to achieve it, we first need to determine the vertex of the quadratic function. The vertex of a quadratic function can be found using the formula x = -b/2a.
Step-by-step explanation:
To find the maximum profit and the number of items that must be sold to achieve it, we first need to determine the vertex of the quadratic function. The vertex of a quadratic function in the form P(x) = ax² + bx + c can be found using the formula x = -b/2a.
In this case, P(x) = -4x² + 16x - 7. So, a = -4, b = 16, and c = -7. Plugging the values into the formula, we have x = -16/(2 * -4) = -16/-8 = 2.
The x-coordinate of the vertex is 2, so the number of items that must be sold to achieve maximum profit is 2.
To find the maximum profit, we can substitute the x-coordinate of the vertex into the quadratic function. P(2) = -4(2)² + 16(2) - 7 = -4(4) + 32 - 7 = -16 + 32 - 7 = 9.
Therefore, the maximum profit is $9.