Higher-order derivative test
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In mathematics, the higher-order derivative test is used to find maxima, minima, and points of inflection for sufficiently differentiable real-valued functions.
The general derivative test for stationary points
Let be a real-valued, sufficient differentiable function on the interval and an integer. If now holds
n is odd and we have a local extremum at c. More precisely:
- is a point of a maximum
- is a point of a minimum
n is even and we have a (local) saddle point at c. More precisely:
- is a strictly decreasing point of inflection
- is a strictly increasing point of inflection
. This analytical test classifies any stationary point of .
The function has all of its derivatives at 0 equal to 0 except for the 8th derivative, which is positive. Thus, by the test, there is a local minimum at 0.
- First derivative test
- Second derivative test
- Second partial derivative test
- Saddle point
- Inflection point
- Stationary point
- Stewart, James (2008). Calculus: Early Transcendentals, 6th ed., Brooks Cole Cengage Learning. ISBN 978-0-495-01166-8
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