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Second derivative test - Multi variable calculus!!

Hi tsr,

What do we do when the second derivative test fails? How do we approach it, and is there a general method to further find whether a critical point is a maximum, minimum or a saddle point?

For example, I'm asked to find all the critical points of the function
f(x,y)=x^2013−y^2013
and determine the nature of the critical points. The critical point that I have found is at (0,0), but I'm unable to determine its nature as the second derivative test fails here.

Any help would be much appreciated
Original post by gn17
Hi tsr,

What do we do when the second derivative test fails? How do we approach it, and is there a general method to further find whether a critical point is a maximum, minimum or a saddle point?

For example, I'm asked to find all the critical points of the function
f(x,y)=x^2013−y^2013
and determine the nature of the critical points. The critical point that I have found is at (0,0), but I'm unable to determine its nature as the second derivative test fails here.

Any help would be much appreciated


When the test fails, it's best to proceed by examining the behaviour of the function in a region local to that point. Alternatively, you could use the extremum test if you know about it.
Reply 2
Original post by Indeterminate
When the test fails, it's best to proceed by examining the behaviour of the function in a region local to that point. Alternatively, you could use the extremum test if you know about it.


Sorry haven't heard about the extremum test.
Original post by gn17
Sorry haven't heard about the extremum test.


In that case, consider how the function behaves in the vicinity of (0,0), i.e on either side.

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