Thread: Extrema test for any dimension

Given,

countinous on a ball, has a critical point on is there a way to determine whether or not this point is a relative extrema?

I am asking this because, there is a foolproof method for single variables called first-derivative test. But for two variables the second-partials test is not foolproof. I wish to know if there is one that is for two and more variables?

there is some sort of way to check it
i think it was with minors of the jacobian of the function

(learned it over a year ago and almost flanked on that course)