Consider a typical "fenced garden" problem, which is easily represented by a quadratic, e.g.,
$a(w)=-10w^{2}+100w$ where $w$ is the width (or the wall, if you prefer).
which peaks at $(5,250)$ and has zeros at $0$ and $10$.
And a similarly typical "box volume with cutouts" problem which is easily represented by a cubic, e.g.,
$v(c)=4c^{3}-12c^{2}+9c$ where $c$ is the length of the cutout.
which peaks at $(0.5,2)$ and has zeros at $0$ and $1.5$.
Now, to the left of each lower bound of the domain of each function lies a non-sensical negative length, and the values of the area and volume functions reflect this by having similarly non-sensical negative area and volume. However, to the right of the upper root they behave differently! Even though the lengths (i.e., the domains: $w$ and $c$) are real and sensible (positive), the quadratic is negative, reflecting that you're over-running the total fence length, and would have to create fence from the air (that is, you have negative fence) in order to make this work. However, the quadratic does a completely different thing, in essence allowing you to create cardboard from the air in exactly the way that the quadratic appears to $not$ permit! And, of course, you can go happily on your way making a larger and larger $positive\ volume$ box, whereas you can't do that with the area! (You can go happily on your way, but you end up with a larger and larger $negative$ non-sense area!)
This may be a philosophical question instead of a mathematical one, and I'm happy to be told so. It also may be a "that's just how it is" thing, like Gabriel's Horn, and, again, I'm happy to be told so. Or maybe there's some interpretation where these both make sense, and if that's the case, I'd love to learn it!
Thanks in advance!