Suppose the cost of producing x items is given by C(x)=1000-x^3, and the revenue made on the sale of x-items is R(x)=100x-10x^2. Find the number of items which serves as a break-even point.

Answer 1

Break even point is when cost equals revenue (i.e. zero profit).
So for
C(x)=R(x), we have
1000-x^3=100x-10x^2
Rearrange to give
-x^3+10x^2-100x+10x^2+1000=0
which solves easily to
x=10