Zenoferox has a very hands-on way to teach his students a math problem: suppose you have a rectangular piece of paper, a*b centimeters on the side. You cut out four x*x squares at each corner, and fold the margins you get to get a box with one face missing. How do you choose x to maximize the value of the box?
For the exact answer in terms of a and b, ask Dr. Math. What I’m thinking of is more about teaching methods for maximum/minimum problems.
The creative exercises, like the box problem or the window problem (maximize the area of a window shaped like a semicircle sitting on a rectangle, given a perimeter a), are a good way of applying principles already learned. I still think the best way to teach the differentiation rule for maximum/minimum problems is to start with a rectangle of a given perimeter and ask to maximize its area. In that case there are enough alternative formulas to help create the intuition that the area is maximized when the rectangle is a square.