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 $n\geq 2$ an integer. If now holds $f'(c)=\cdots =f^{(n-1)}(c)=0\quad {\text{and}}\quad f^{(n)}(c)\,\not =0$

then, either

n is even and

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

or

n is odd and

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

.

See also

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