Calculus of Variation: Extremum & Further Variances

[LagrangeEuler]

If for some functional $I$, $δI=0$ where $δ$ is symbol for variation functional has extremum. For $δ^2I>0$ it is minimum, and for $\delta^2I>0$ it is maximum. What if
$δI=δ^2I=0$. Then I must go with finding further variations. And if $δ^3I>0$ is then that minimum? Or what?

 

[muzialis]

Finding further variations is useless from this point of view.
The stationarity of the functional, i.e. δI=0 , occurs for maxima, minima and saddles.

 

[LagrangeEuler]

So then how I could know? Is it minimum or maximum?

 

[muzialis]

Here you will find a better explanation than I could give on sufficient and necessary conditions for minima http://www.math.utah.edu/~cherk/teach/12calcvar/sec-var.pdf
If you have the book "introduction to Calculus of Variations" by Fox you will find there a thorough discussion of the second variation: yes further variations are to be computed.
I really do apologise for my previous reply which was wildly inaccurate due to a misunderstanding of mine.