Question

A company estimates its total profit to be modeled by the function p(x) = x3 - 4x2 - 1, where p is in hundreds of dollars and x is the number of years elapsed since the start-up year. Do you expect that the company will break even?

Answer 1

To find out if the company will break even, we look for the roots of the profit function p(x) = x^3 - 4x^2 - 1. The presence of at least one real root indicates a break-even point.

Step-by-step explanation:

The student's question regards whether a company will break even based on the profit function p(x) = x^3 - 4x^2 - 1. To ascertain when the company breaks even, we need to determine when the profit equals zero, which involves finding the roots of the profit function. If the profit function has at least one real root, this indicates that the company will reach a break-even point at some time.

However, to determine the exact times when this happens, we would need to solve the cubic equation, usually by finding real factors or applying methods such as synthetic division, or numerical methods if the roots cannot be easily factored.

While the process can be elaborate, the mere presence of a real root to the profit function suggests that the company is expected to break even at some point.