An open-top box is made by cutting squares of side length $x$ from the corners of a 12-inch square sheet and folding up the sides. Which value of $x$ maximizes the volume of the box?

Core Concept

$V=x(12-2x)^2$, and $V'=12(x-2)(x-6)$ has critical points $x=2$ and $x=6$. Since $V(2)=128$ and $V(6)=0$, the maximum volume occurs at $x=2$.