Final answer:
The profit-maximizing choice for a perfectly competitive firm occurs at the level of output where marginal revenue is equal to marginal cost. In this case, the x-values that make the profit equal to zero are x = 0 and x = -6000.
Step-by-step explanation:
The profit-maximizing choice for a perfectly competitive firm occurs at the level of output where marginal revenue is equal to marginal cost. This means that the profit is maximized when the rate at which revenue increases (marginal revenue) is equal to the rate at which cost increases (marginal cost). In this case, we can use the given functions:
c(x)=2x and R(x)=8x-0.001x²
To find the x-values where profit is zero, we need to find the values of x where the total revenue (R(x)) equals the total cost (c(x)). So, set R(x) equal to c(x) and solve for x:
8x-0.001x² = 2x
0.001x² + 6x = 0
x(0.001x + 6) = 0
x = 0 or x = -6000
The x-values that make the profit equal to zero are x = 0 and x = -6000.