Would a parabolic cup stand upright? Imagine a parabolic cup which is formed by rotating a parabola $y=ax^2+bx+c$ about its axis of symmetry to form a plastic cup.  Question: is there any height of such a cup along with some amount of water poured into it such that the cup would stand upright?
I think it's clear the answer is "no" when we have a narrow parabola (e.g., $y=10x^2$) but perhaps there is a way to make this work if the parabola is broad enough?  And is there a rigorous way to prove that it would?
 A: What happens if we topple the cup ever so slightly (on a plane table)? 

Consider a thin layer of water at some specific height.
Ignoring its thickness, it is just the intersection of a plane with a rotational paraboloid, which happens to always be a circular disk (i.e., not only with planes perpendicular to the rotational axis!).
The centre of gravity of this thin water disk is at the centre of the circle, i.e., the midpoint of the line segment in the above cross section image.
For the example layer in my picture as well as for any sufficiently high layer, it is clear that this cog is to the left of the point where the cup touches the table. This would tend to amplify the toppling and so the cup will topple over completely and spill out all the water - what a mess! But perhaps this is not the case for very low layers? And perhaps with only a little bit of water, there is actually a chance that there is more water to the right, thus making the vertical position a stable equilibrum?
To compute this, tilt your head that the parabola looks upright again, $$y=x^2,$$ and the tangent table is skew instead. To make it touch at $(a,a^2)$, we write the tangent as
$$ y=2ax-a^2.$$
For $h>0$, where does the secant
$$ y=2ax-a^2+h$$
intersect the parabola? Of course, the $x$-coordinates are the roots of
$$ x^2-2ax+a^2-h, $$
which are
$$a\pm\sqrt{\cdots},$$
i.e., their midpoints is at $$(a,a+h),$$
i.e. on the line through the touch point and parallel to the axis of symmetry.
Back to our image above, this means that the cog of every layer of water is to the left of the touch point. The equilibrum is instable.
