Unexpected saddle points from Wolfram|Alpha?

Questions
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1

My problem is shown in the picture:

saddlepoints.png
saddlepoints.png995Ă—489 34.7 KB

I would have expected that in (0,0,0) there was a saddle point, instead Wolfram|Alpha contradicts me. Could I know why? What definition are you referring to? Thanks!

2

In Mathematica:

f[x_, y_, z_] := x^2 - y^2 + z^3;
SaddlePoint = Solve[{D[f[x, y, z], x] == 0, D[f[x, y, z], y] == 0, D[f[x, y, z], z] == 0}, {x, y, z}]
(*{{x -> 0, y -> 0, z -> 0}, {x -> 0, y -> 0, z -> 0}}*)

Det[( {
    {D[f[x, y, z], {x, 2}], D[D[f[x, y, z], x], y], 
     D[D[f[x, y, z], x], z]},
    {D[D[f[x, y, z], y], x], D[f[x, y, z], {y, 2}], 
     D[D[f[x, y, z], y], z]},
    {D[D[f[x, y, z], z], x], D[D[f[x, y, z], z], y], 
     D[f[x, y, z], {z, 2}]}
   } )] /. SaddlePoint[[1]]
  (*0*)

   (* Determinant gives a zero,(the test gives no information)  then we do not learn whether we have a local max, a local min, or a saddle point at (0,0,0)*)

Looks like WolframAlfa are correct.

3

The definition given by Wikipedia is

In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function

The origin is a stationary point of x^2 - y^2 + z^3, in the direction of the x axis it is a minimum, in the direction of the y axis it is a maximum, and in the direction of the z axis it is an inflection. I would conclude that it is a saddle point in this sense.

The matrix of the second derivatives has zero determinant, but it clearly has -2,2,0 as eigenvalues. This should be sufficient to decide that it is a saddle.

Maybe Wolfram|Alpha requires the saddle points to have all nonzero eigenvalues.

4

Thanks to the responses received I tried with:

aaaa.png
aaaa.png1000Ă—428 30.7 KB

where also in this case the Hessian is null, but the saddle point is identified.

I conclude that the previous message “no saddle points found” is proper to interpret that the Wolfram | Alpha algorithms have not been able to identify any saddle points, but this does not imply that there cannot be.

Thank you all!

5

Or maybe the algorithm is designed for functions of two variables, and it gives meaningless results in three variables.

6

Again thanks to the inputs I received from the answers above I tried with:

9911aaaa.png
9911aaaa.png1002Ă—474 33 KB

and again Wolfram|Alpha was unable to identify a saddle point.

I could have thought about it before, my fault, sorry for wasting your time.