Applied Max / Min Problems

We are spending this week studying applied max / min problems in Chapter 3 of Stewart’s Calculus book (we are using Edition 3). The chapter is a mix of theory (Rolle’s theorem, the mean value theorem), ideas about finding max and mins of functions, and finally applied max / min problems. We discussed the theory a bit, and I’m planning on circling back on the theory later in the year, we’ve already informally covered ideas about max and mins, so I wanted my son to focus mainly on the applied max / min problems.

The problem he’s presenting here goes like this:

Find the largest rectangle that can be inscribed in an equilateral triangle if one of the sides is on the base of the triangle.

So, a pretty standard problem, I guess, requiring a bit of geometry.

Here’s his solution:

After he presented his solution I gave a few comments to help him understand

(i) a slightly different way to approach the geometry (this is closer to a “fun fact” rather than an important piece of learning calculus).

(ii) one idea about finding max / mins of functions that might help him simplify some of his calculations, and

(iii) using the second derivative to tell if you have a max or a min