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 $x=2$.
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 $x=2$.