To get better acquainted with "plot3d," I browsed through the help documentation. I was surprised to discover not only "Polyhedraplot," but—via "Polyhedra Sets" and "DualSet"—the very tools needed to tackle a classic subject: The duality of Platonic solids.
This reminded me of the following puzzle:
Consider the unit cube (edge ​​length = 1) and, situated within it, its dual polyhedron (the octahedron). As is well known, the dual polyhedron of the octahedron is, in turn, a cube. Now, let us continue this dual construction indefinitely, starting from the unit cube (cube containing octahedron containing cube containing octahedron...). This generates a sequence of nested polyhedra—alternating between cubes and octahedra.
1.) For each of the infinite subsequences—that of the cubes and that of the octahedra—calculate the sum of all volumes and surface areas.
2.) What is the maximum volume an octahedron (regardless of its orientation) can have while being completely contained within the unit cube?

In the attached file, I would like to practice calculating distances between the exact and approximate functions using the "Minimize" function. However, the approach using the integral metric according to (4) produces an error message. I am unable to correct this and would appreciate some advice.test.mw

Download test.mw

...and starting values. I want to reproduce the maximum in Maple using the solution structure in the attached file, similar to the old Mathcad method. Instead of using the Euler-Lagrange equation, I want to separately enter the objective function, then the boundary conditions, and finally the starting values ​​for an iteration (as was done in the old Mathcad solution block). I entered the latter into the "maximize" command using the Maple help text. But nothing is being calculated. What am I doing wrong? As I said, I only want a numerically generated approximate solution.test.mw

Download test.mw

With thanks to @sand and @janhardo for their help in solving a variational problem using Maple, I'm presenting a puzzle worth remembering. It was originally published in an IBM competition over 20 years ago under simpler conditions than those presented here, and later posed in a more challenging form in private circles:

A farmer wants to buy a section of a large meadow/pasture for his sheep. He likes a lone, large oak tree on the meadow. This tree is to be the starting and ending point of the fence around the desired section. The fence is exactly 100 meters long. The farmer creates a sketch of the situation. He places the oak tree at the origin of a Cartesian coordinate system with a positive y-axis pointing north and sketches the estimated location of the desired meadow section east of the oak tree. The seller then stipulates that the price per square meter increases linearly eastward by a factor of 1/m³. An agreement is reached, and the farmer now considers the optimal route for the fence to maximize the return on his investment.

The fence route without corners and the value of the purchased meadow must be calculated.

In the attached file, I would like to evaluate the integral according to (16) and the minimum according to (17). I would appreciate your help. The goal is to calculate the coefficients a and b.

Euler_eq.mw