$^3$
of a box (without a lid) that can be obtained by cutting out a square of side $x$
from each corner and then folding the flaps up?

What are the dimensions of the box formed in terms of
$x$
? 
What is the function that defines the volume of the box?

How may we find the maximum point of that function?

Cutting sides of length
$x$
and folding up the flaps results in a prism of height$x$
and a square base of side$102x$
. The volume is therefore given by$V(x)=x(102x)^2$
. Differentiate this to find the stationary points:$$V'(x)=(102x)^24x(102x)=4(x5)(3x5).$$
So
$V'(x)=0$
at$x=5$
and$x=\frac{5}{3}$
. Substitute back in to find which of them gives the higher value (that’s quicker than doing second derivatives), to get that$\frac{5}{3}$
gives the maximum. Hence our final answer is$V\left(\frac{5}{3}\right)=\frac{2000}{27}.$