What to do when second partial derivative test is inconclusive

[Zoe-b]

Homework Statement

I have two problems where there is a critical point of f(x,y) at (0,0), but the second derivatives and mixed second derivative are all zero. The second partial derivative test is therefore inconclusive- all the information I can find online/in my notes just says it is inconclusive and doesn't offer an alternative method. If anyone could give me a link or even just tell me what I should be googling that would be really helpful! Thanks.

Homework Equations

The Attempt at a Solution

I vaguely think I could show the critical point was a saddle point if I could show along one curve f(x,y) is positive as (x,y) -> (0,0) and along another curve f(x,y) is negative as (x,y) -> (0,0) but I have no theorem that states this.

 

[lanedance]

so in the single variable case you would consider the 3rd derivative

if you consider a passing along a line eg x or y axis, or y=x, as you say you could either consider the 3rd derivative, however whilst you may be able to show a saddle point, it won't be able to confirm an extremum

 

[lanedance]

so what was the function?

 

[Zoe-b]

The first one is f(x,y) = (x^2)y + x(y^2)

but I was really after ideas for a method not the solution.

 

[lanedance]

well i don't know any cook book methods, but you could have a think about what happens to the higher order derivatives, or the previous idea

however for this case I would consider the line y=x (or the axes)