Q.An open box is to be made from a square piece of cardboard of side $18\ \text{cm}$ by cutting equal squares of side $x$ from each corner and folding up the sides. Find the value of $x$ for which the volume of the box is maximum, and find this maximum volume.

A real-life optimization problem -- maximizing an enclosed area for a given perimeter, minimizing the material for a container of given volume, maximizing a product for a given sum, and so on -- is solved by expressing the quantity $Q$ to be optimized as a function of a single variable, using the problem's own constraint to eliminate every other variable, then applying the ordinary critical-point method: differentiate, solve $Q'(x)=0$, and classify each critical point with the first or second derivative test. When the variable is restricted to a closed interval by the physical setup (e.g. the side cut from a fixed-size sheet), the values of $Q$ at the interval's endpoints must also be checked against the interior critical point, since an endpoint can sometimes give a more extreme value than any interior turning point. The final answer should always be reported …