Critical Points

Flat spots come in three flavours, and one determinant tells them apart. Watch a peak and a saddle collide and annihilate.

Landscape

λ 1.00

minimum — bowl, D > 0 and f_xx > 0
maximum — dome, D > 0 and f_xx < 0
saddle — D < 0

flat spots found: —
what they are: —

Slide λ down to zero and watch two of them meet. At the instant they touch, D is zero and the test says nothing.

Slide the parameter until the two critical points meet.

What you're looking at

A flat spot is any place the gradient vanishes — no uphill direction, no downhill one. In one variable that leaves two possibilities and the second derivative settles it. In two variables there is a third: the surface can rise one way and fall the other, and that's a saddle.

So a single second derivative isn't enough; you need all of them, arranged into the discriminant D = f_xxf_yy − f_xy². When D is negative you have a saddle, full stop. Only once D is positive does the sign of f_xx get to say whether you're in a bowl or under a dome.

That cross term matters more than it looks. A surface can curve upward along both axes and still be a saddle, because the fall-off hides in a diagonal direction that neither axis sees. The determinant catches it; checking f_xx and f_yy alone does not.

And when D = 0 the test genuinely has no answer — the quadratic approximation is flat in some direction and whatever decides the behaviour lives in higher-order terms. Slide λ to the moment two flat spots collide and you're looking straight at that case: they merge, annihilate, and leave nothing behind.