How do you do this function problem? I am so lost.

Question

A company determine that its total profit is given by the function P(x). The company makes a profit for those nonnegative values of x for which p(x)>0. The company loses money for those nonnegative values of x for which p(x) <0. Find the value of x for which the company makes a profit and loses money. p(x)=(x+1)(x-2)² (x-4)

Answer 1

Basically, what you want to find is the x values in which p(x) is negative and the values for which p(x) is positive. In other words, you want to find all of the x values that have a y value of 0 and then determine if the function is increasing or decreasing at that value. We know this to be called x-intercepts or root values. 

 

The root values for this equation is x = -1, 2, and 4. We know that we have to have non negative values so x = -1 is automatically thrown out. Lets look at x = 2.

 

If you were to evaluate p(1.9) and p(2.1) you would see that both of these value are negative. This means that at x = 2 our profit does not turn from negative to positive. Our only hope is at x = 4

 

If you were to evaluate p(3.9) and p(4.1) you would see that the values turn from negative to positive. x = 4 is considered the break-even point since we have a profit of 0. Every point after 4 is a positive which is where any company would want their profit to be.