What is the extremum when first and second variations vanish?

[LagrangeEuler]

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

 

[voko]

I suggest you ask the same question in a simpler situation: ordinary single-variable function.

 

[dauto]

Use the first non-vanishing derivative. If it is an odd derivative, than you have and inflection point, otherwise it will be either a maximum or a minimum depending on its sign