Higher-order derivative test

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

Let f be a real-valued, sufficient differentiable function on the interval I \subset \R, \; c \in I and n \ge 1 an integer. If now holds f'(c)=\cdots=f^{(n)}(c)=0\quad \text{and}\quad f^{(n+1)}(c)\,\not= 0

then, either

n is odd and we have a local extremum at c. More precisely:

  1. f^{(n+1)}(c)<0 \Rightarrow c is a point of a maximum
  2. f^{(n+1)}(c)>0 \Rightarrow c is a point of a minimum


n is even and we have a (local) saddle point at c. More precisely:

  1. f^{(n+1)}(c)<0 \Rightarrow c is a strictly decreasing point of inflection
  2. f^{(n+1)}(c)>0 \Rightarrow c is a strictly increasing point of inflection

. This analytical test classifies any stationary point of f.


The function x^8 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.

See also[edit]