Question

The total profit function P(x) for a company producing x thousand units is given by `P(x) = -3x^2 + 36x - 81`. How do you find the values of x for which the company makes a profit?

Answer 1

I found: `3<` `x` `<9` (in thousand units)

Explanation:

You can try plotting your function considering that it represents a parabola:

You can see that producing zero units (`x=0`, i.e., the company is idle) you get a loss, `P(0)=-81` this is because you have to pay fixed costs, say, taxes, rent...etc.
There is a region where `P(x)>0` and you have a profit, where:
`-3x^2+36x-81>0` that gives you (using the Quadratic Formula):
`3<` `x` `<9` (in thousand units) that described the red shaded region int the graph where `P(x)>0`.