You have a card of 10cm by 10cm. What is the largest volume in cm$^3$ 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 $10-2x$. The volume is therefore given by $V(x)=x(10-2x)^2$. Differentiate this to find the stationary points:

$$V'(x)=(10-2x)^2-4x(10-2x)=4(x-5)(3x-5).$$

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 $V\left(\frac{5}{3}\right)=\frac{2000}{27}.$