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[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

or

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.

See also[edit]