c(x)=5x+49, r(x)=-x²+21x.
If p(x) is the profit, p=r-c=-x²+16x-49.
This can be written:
-49-(x²-16x)=-49-(x²-16x+64-64),
which is the same as:
-49-(x-8)²+64=15-(x-8)².
The maximum value of this expression is 15, when x=8.
So 15 is the maximum profit.
Breakeven is when r=c, so p=0, that is:
(x-8)²=15, x-8=±√15=3.87 approx.
Therefore, x=8±3.87=11.87 or 4.13.
But x must be a whole number, so if x=11, p(11)=6, and if x=12, p(12)=-1.
If x=4, p(4)=-1 and if x=5, p(5)=6.
So there is no exact breakeven point, but is close to x=4 and 12. Therefore the interval when a strict profit is being made (p>0), is between x=5 and 11, when the profit is between 6 and 15.