Higher-order derivative test

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 $f$ 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 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 $f$.

Example

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

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

References

• Stewart, James (2008). Calculus: Early Transcendentals, 6th ed., Brooks Cole Cengage Learning. ISBN 978-0-495-01166-8