A company determines that its total profit is given by the function P ( x ) equals negative 3 x squared + 567 x - 4860. 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 values of x for which the company makes a profit and loses money.

As we can see, the profit function, P (x) is a quadratic function. It means it is continuous, and so we can apply the theorem which states that if the function is continuous and there is a value which is positve and another one that is negative, there is at least 1 root in the middle. Also, we need to consider the characteristics of the quadratic function: two roots, a parabola shape in the graph and a vertex in which both branches are equal at each side of the vertex.

The quadratic function has this general formula: a*x^2+b*x+c . So in this case, we have a=-3 b=567 and c=-4860

The formula for calculating vertex is -b/(2*a) . In this case, it will be -567/(2*(-3) and result is 94.5.

The way of seeing if the parabola goes up or down is by seeing a. If a<0, it goes down and if a>0 it goes up. a=-3 so it goes down and has this shape: ∩

So the vertex is actually the maximum. We have profits increasing up to that point and then decreasing. Profits for that point, i.e. P(94.5)=21930.75

When will profits be 0? We have to see the x values that makes P (x)=0 which means to find the roots.

If we use the formula for roots (see attached) we will see that both roots are X1=9 and X2=180

If we take a point in X less than 9, for example 8, we see that P(x) is negative (-516 if x=8). So if we apply the theorem I mentioned at the beginning, we can see that in the interval (8;94.5) the image of the first value is negative and the second one is positive, which means there is a root, which is 9. We can conclude that every value after 9 and until 94.5 is a positive one, so company will have a profit.

From 94.5, the profits will fall as X rises (because, as I said, 94.5 is the maximum). Profits (P(x)) will still be positive, but decreasing until it hits the other root, 180. If we take X values more than 180, we will find XP(x) to be negative. For example P (181)=-516. So if we take the same theorem we applied before and the same reasoning, we can conclued that there is a root in 180 which is the poin in which the function P(x) stops being positive and starts being negative.

Hence, the interval in which P(x) will be positive will be from x=9 up to X=180 and will be negative if X is less than 9 or more than 180