Do have an example of a function f : R 2 → R that has a critical point that is neither a saddle point nor a local (or global) extremum?

Question

Do have an example of a function   f  :      R    2    →  R that has a critical point that is neither a saddle point nor a local (or global) extremum?

Gillian Howell · Accepted Answer

You may look at   f  (  x  ,  y  )  =      x    2    +      y    3    −  3  x      y    2  Then there are   2 stationary points :   (  0  ,  0  ) and   (  1      /    6  ,  1      /    3  )You can check that   (  1      /    6  ,  1      /    3  ) is actually a saddle point. However, for   (  0  ,  0  ), we have   f  (  x  ,  0  )  =      x    2   and   f  (  0  ,  y  )  =      y    3  On the   x-line it&#039;d be a minimum. On the y-line it would be a saddle point. This point is neither an extremum, or a saddle-point.

Do have an example of a function f/R^2 rarr R that has a critical point that is neither a saddle point nor a local (or global) extremum?

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