Read the result
Find the first derivative value that is not zero. Its order and sign control the local shape.
Optimisation
Higher-Derivative Stationary Point Classifier
Classify a stationary point when the second derivative test gives zero.
Formula
first non-zero derivative: even positive -> min, even negative -> max, odd -> stationary inflection
Result
Worked substitution
The first non-zero derivative controls the local shape near the candidate point.
Where it helps
Use it for flat stationary points such as x^4 or x^3, where the ordinary test is inconclusive.
Common slip
If the first non-zero derivative is odd, do not label the point as a maximum or minimum.
Try it
Set f''''(a) positive, then negative, and watch the classification swap between minimum and maximum.