A common example to introduce quadratic functions is to ask for a rectangle with the largest area when the perimeter is given and you are allowed to use one additional edge that does not count towards the perimeter ("Building a fence for an rectangular enclosure next to an existing wall").
What happens if you lift the "rectangular" limitation and allow arbitrarily shaped shapes?
I found out that a semicircle is better than the best rectangle, but I have no idea how to check for other shapes with curved boundaries.
So, the question is:
What is the largest possible area that one can get with a shape that has one straight edge and an otherwise arbitrarily shaped boundary of length $x$?
This is a problem from antiquity calles "Dido's Problem" or "Dido's isoperimetric Problem", the solution is a semi-circle.
See for example here or here.
It's clear that the resulting shape has to be convex, because otherwise one can just flip an inside bulge to an outside bulge, thereby inclreasing the area but keeping the perimeter unchanged. See here for a description.
Likewise, and under the assumption that the curve is smooth, when the curve has an inside angle of $\phi > 90°$ with the line, a portion of the curve can be flipped inside out such that the area increases, the length of the curve stays the same, and $\phi < 90°$ after the flip. Thus $\phi\leqslant 90°$.
The technique used to solve the prolbem is calculus of variations, where the problem is stated a bit differently: One assumes that the curve starts and ends at fixed points $a$ and $b$ of the line, and one then shows that the curve is part of a circular arc.