Looks like both methods give the same solution to me….
I agree that my counterexample using √y is broken and does not
illustrate the point I wanted to make.
However, you used /calculus/ to show that it was broken.
In any case, the overarching point remains: In a polynomial
over x, when the coefficients are themselves functions of x,
it is not self-evident that the place where the discriminant
goes to zero is the minimum.
There are numerous ways of showing that the result is true:
a) by using calculus
b) by expanding things to lowest order
c) or by an even simpler argument:
The quadratic equation says there are two values of x corresponding
to the same y. If you look at the graph of the function, the only
place where this happens, for two x values close together, is near
the minimum. So the place where the discriminant goes to zero
must be the minimum.
That is more simple and more geometrical, in keeping with the
goals of the question as originally asked.
IMHO just saying «evidently» isn't good enough. Just because
something is true doesn't make it self-evident.
Speaking for myself, the algebraic approach, i.e. expanding things to
first order, is super-easy. I can do that blindfolded. And it is
guaranteed to work. For me, it is easier than considering the geometry
of the situation, which would require actual thinking.
OTOH the geometrical approach is entirely correct. Anybody who wants
to take that route is welcome to do so. Still, though, this step is
not self-evident and ought not be skipped.
==========
If you want an example of where you really need to be careful about
such things, here's a well-known example:
1) Tell me, where is the peak of the black-body spectrum, when
you plot it as a function of the frequency f?
2) Tell me, where is the peak of the black-body spectrum, when
you plot it as a function of the wavelength λ?
Those two questions give incompatible answers. That is, the answers
are not related by λ = c f.
The point remains that in general, the location of the minimum is *not*
invariant with respect to a change of variable. In the ramp problem
the change of variable is harmless, but this is not self-evident.