The primary difference between case 1 and case 3 is that the former is a polynomial, whereas the latter is not.

As per Mathematica documentation for `Minimize`

, if the function to be minimzed and the constraints (if any) for minimization happen to be polynomials, then Minimze would give us the global minimum. A quick 3D plot for 1 reveals that it has no global minimum, which explains why `Minimze`

demonstrates the behaviour it does.

In fact, 1 describes a saddle surface.

In case of 3, however, it can be clearly understood that the function descends from $infty$ as `Abs(x)`

grows larger, on either side of the $y$-axis, and as `Abs(x)`

$ rightarrow infty $ on either side, `y`

$ rightarrow 0 $ asymptotically. There’s no global minimum again, in-fact, no stationary points at all (contrast this with saddle/inflection points of a curve, where the univariate function is stationary, even if it’s not an extremum).

So, in both cases, the output obtained is what one should expect, going by the behaviour of the function, and the Mathematica documentation for `Minimize`

.