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?