Higher-order derivative test

In mathematics, the higher-order derivative test is used to find maxima, minima, and points of inflection for sufficiently differentiable functions.

The test

Let $f$ be a real-valued, sufficient differentiable function on the interval $I\subset \mathbb {R} ,\;c\in I$ and $n\geq 1$ 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 even and we have a local extremum at c , preciser:

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 odd and we have a (local) saddle point at c, preciser:

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

. E.g. this analytical test classifies any stationary point of $f$.

See also

• Extremum
• First derivative test
• Second derivative test
• Hessian_matrix#Second_derivative_test
• Saddle point
• Inflection point
• Saddle-point method
• Stationary point