Extrema in Several Variables: A different way?

[nth.gol]

Hi,

In high school, I was shown an unconventional but quicker way to find max/mins. I'm not sure how common it is but we did it because we learned to curve sketch without calculus first.

Take f'(x) =0, and solve for the roots. Construct a number line and place all roots on the number line. Alternate + - from the right, unless there is a negative out front, and don't change signs around squared roots.

From here you extrapolate max mins based on sign. This is all nice and dandy compared to the first derivative test.

In several variables, however, I am currently being taught to use the partial second derivative or Determinant test.
Is there a better way? A quicker one like this? I understand geometrically the implications of the second partial test, and the cases where the pure and mixed partials affect the type of extrema. But is there a similar test to that in single variable?

Thank you

 

[HallsofIvy]

That is, in fact, what is known as the "first derivative test". It is typically given in most Calculus texts just before the "second derivative test". It is simpler than the second derivative test in that you do not need to find the second derivative. However, it also requires that you be able to find the sign of that first derivative on intervals rather than at individual points as with the second derivative test. Which is easier really depends upon the function.

 

[nth.gol]

That is, in fact, what is known as the "first derivative test". It is typically given in most Calculus texts just before the "second derivative test". It is simpler than the second derivative test in that you do not need to find the second derivative. However, it also requires that you be able to find the sign of that first derivative on intervals rather than at individual points as with the second derivative test. Which is easier really depends upon the function.

Im not sure you really answered my question, but thanks for the input.
Is there another test in several variables?