A company’s profit is described by the equation

P(x) = – 5x^2 + 300x + 15,000

where x is the price in dollars that the company charges for its product. What should the company charge for the product to generate the maximum profit?
a.) $20
b.) $30
c.) $50
d.) $60

How to solve? Please explain.

the vertex is at x = -b/2a = -300/-10 = 30

That's the equation of a parabola opening downward, so the vertex will be the highest point on the parabola, and its x-coordinate will be the value of x producing that highest point. The formula for the x-coordinate of the vertex is -b/(2a) = -(300)/[2(-5)] = (-300)/(-10) = 30. Answer: b

Thanks.

To find the price that would generate the maximum profit, we need to find the x-value (price) that corresponds to the maximum value of the profit function P(x) = -5x^2 + 300x + 15,000.

To do this, we can use calculus by taking the derivative of P(x) and setting it equal to zero to find the critical points. The critical point(s) will help us identify whether the profit is increasing or decreasing around that point.

1. Take the derivative of P(x) with respect to x:
P'(x) = -10x + 300

2. Set P'(x) equal to zero and solve for x:
-10x + 300 = 0
-10x = -300
x = -300/-10
x = 30

The critical point is at x = 30, which represents the price that corresponds to the maximum profit.

Now, we need to determine whether this critical point is a maximum or minimum by examining the concavity of the function. This can be done by taking the second derivative of P(x).

3. Take the derivative of P'(x) with respect to x:
P''(x) = -10

Since the second derivative P''(x) is constant (-10), and it is negative, this indicates that the function is concave down, which implies that the critical point x = 30 is a maximum.

Therefore, the company should charge $30 for its product in order to generate the maximum profit.

Therefore, the correct answer is b.) $30.