# Identifying local maximum, local minimum and saddle point.

## Main Question or Discussion Point

Indicate whether you think it is a local maximum, local minimum, saddle point, or none of these? My solution:

Point P = Local Max
Point Q = Local Min
Point R = None

I got a 75% for first attempt, so one answer is not correct and I am not sure which one isn't.

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mfb
Mentor
The circle around Q is confusing. If it is really +1, it should have a zero ring around it. If it is -1 (what I would expect), where is the minus sign?
Same problem with the circle to the right of R, just with reversed signs.

Either R or Q give the issue, which one depends on those sign problems.

Ben Niehoff
Gold Member
The circle around Q is confusing. If it is really +1, it should have a zero ring around it. If it is -1 (what I would expect), where is the minus sign?
Same problem with the circle to the right of R, just with reversed signs.

Either R or Q give the issue, which one depends on those sign problems.
Actually the problem with Q is bigger than that. The 0 contour is marked in the picture: it is the pair of diagonal lines going through S. Given the 1 contour around Q, where should the 0 contour go? There must be one between -1 and 1 (unless the function is discontinuous on a ring around Q, in which case the problem is evil). But if the 0 contour is a ring around Q, then what to make of the 0 contour that is already drawn going through S? Either these contours must join up (possibly at R?), or there must be an additional local min/max somewhere between Q and the origin. Either way, it seems that important information has been omitted from the plot.

mfb
Mentor
The 0 contour is marked in the picture: it is the pair of diagonal lines going through S.
This cannot be the whole 0 contour for a continuous function. There is a path from a -1 to a +1 contour which has to have a 0 somewhere.
But if the 0 contour is a ring around Q, then what to make of the 0 contour that is already drawn going through S?
Where is the problem? (x^2-y^2)^2*((x-1)^2+y^2-0.1) has this type of contours.
or there must be an additional local min/max somewhere between Q and the origin.
It does not have to, but even if it has, where is the problem?
Either way, it seems that important information has been omitted from the plot.
Certainly, as there is no 0 contour or the contour labels are wrong.