Higher derivative test

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Statement

What this test is for

The term higher derivative test or higher derivative tests is used for a slight modification of the second derivative test that is used to determine whether a critical point for a function is a point of local maximum, local minimum, or neither. The higher derivative test can help resolve some of the inconclusive cases of the second derivative test.

What the test says

Suppose is a function and is a point in the interior of the domain of . Suppose is a critical point for of the type where . Suppose further that there exists a natural number such that:

  • The derivative of at , namely , exists and is nonzero.
  • All lower order derivatives of at , starting onward, are zero. In other words, .

In other words, the first nonzero-valued derivative of at .

Then, we have the following:

Case on parity of Case on sign of Conclusion about at
even negative strict local maximum (mimics the second derivative test where )
even positive strict local minimum (mimics the second derivative test where )
odd negative neither local maximum nor local minimum. The point is a point of decrease for the function. In fact, it is also a point of inflection, though that is not of relevance here.
odd positive neither local maximum nor local minimum. The point is a point of increase for the function. In fact, it is also a point of inflection, though that is not of relevance here.

Related tests

Strength of the test

Relation with second derivative test

The test contains the second derivative test under the subcase . Further, in case the second derivative test is inconclusive, it is possible that the higher derivative test would still be conclusive. So, we can think of it as a modification of the second derivative test that helps address some of the inadequacies of the second derivative test.

When is the test conclusive and inconclusive?

Situations when the test is inconclusive

What problem do we run into? What kind of trouble do we have? Can we use the first derivative test?
For critical points that are of the does not exist type, the test cannot be applied. The test doesn't make sense We may or may not be able to use the first derivative test.
The higher derivatives cease being defined before they get a chance to become zero. For instance, if and does not exist, the test is inconclusive. The test crucially relies on determining the position and sign of the first nonzero derivative. If the higher derivative ceases to exist before it becomes nonzero, the test cannot be used. We may or may not be able to use the first derivative test.
All higher derivatives at the point are zero. The test crucially relies on determining the position and sign of the first nonzero derivative. If all higher derivatives are zero, we cannot use the test. We may or may not be able to use the first derivative test.

Conditions for the test to be conclusive