We want to make a square-based, open-top box from a square piece of paper with a side length of $20 \mathrm{~cm}$ by cutting out congruent squares from each of the four corners of the paper and folding up the resulting "flaps". What is the maximum possible volume of the box?
An extremum problem can be solved, after introducing an appropriate variable, either by using the inequality between the arithmetic and geometric means, or by using differential calculus.

The maximum volume is $\frac{1}{4} \cdot\left(\frac{40}{3}\right)^{3} \mathrm{~cm}^{3}$, and this can be achieved when the side of the cut-out squares is $\frac{10}{3} \mathrm{~cm}$. Many people incorrectly concluded the maximum based solely on the zero of the derivative.