Example 2

We consider the revenue function $R(x)=10x$ and the cost function $C(x)=(4000x)^{\frac{1}{3}}$ and we determine the break-even points.

This implies the following:
$$\begin{align} R(x)  = C(x) & \Leftrightarrow 10x=(4000x)^{\frac{1}{3}}\\ & \Leftrightarrow (10x)^3=4000x\\ & \Leftrightarrow 1000x^3=4000x\\ & \Leftrightarrow 1000x^3-4000x=0\\ & \Leftrightarrow 1000x(x^2-4)=0\\ & \Leftrightarrow x=0 \mbox { or } x=-2 \mbox{ or } x=2. \end{align}$$

Due to the non-negativity of production levels, the break-even points are $x=0$ and $x=2$.