Multivariable Calculus Clarification Question

I just started to learn about extrema in calc III, and I just got to the second-partials test. While I understand that this is akin to the second derivative test from calc I, is there no such thing as the first derivative test in multi? Like, can I use multiple sign charts for the components of the gradient to see whether or not it is an extrema or not?

Question

Multivariable Calculus Clarification Question

I just started to learn about extrema in calc III, and I just got to the second-partials test. While I understand that this is akin to the second derivative test from calc I, is there no such thing as the first derivative test in multi? Like, can I use multiple sign charts for the components of the gradient to see whether or not it is an extrema or not?

anonymous · Answer

I don't think there is 1st derivative test for multi-variable functions

anonymous · Answer

Do you think my method at the end there is necessarily wrong though? I can't think of a counterexample for it.

anonymous · Answer

I guess you could try. If there was such method, it would have been written in calculus books already.

anonymous · Answer

Yeah, I guess. I'll just keep on testing it on other practice questions until I hit a snag. If anyone can actual find/think of a case where it doesn't work, that'd be great though!