Editing Derivative test (section)

===Higher-order derivative test===
The ''higher-order derivative test'' or ''general derivative test'' is able to determine whether a function's critical points are maxima, minima, or points of inflection for a wider variety of functions than the second-order derivative test. As shown below, the second-derivative test is mathematically identical to the special case of ''n''&nbsp;=&thinsp;1 in the higher-order derivative test.

Let ''f'' be a real-valued, sufficiently differentiable function on an interval <math>I \subset \R</math>, let <math>c \in I</math>, and let <math>n \ge 1</math> be a [[natural number]]. Also let all the derivatives of ''f'' at ''c'' be zero up to and including the ''n''-th derivative, but with the (''n''&nbsp;+&thinsp;1)th derivative being non-zero:

: <math>f'(c) = \cdots =f^{(n)}(c) = 0\quad \text{and}\quad f^{(n+1)}(c) \ne 0.</math>

There are four possibilities, the first two cases where ''c'' is an extremum, the second two where ''c'' is a (local) saddle point:
* If ''(n+1)'' is [[parity (mathematics)|even]] and <math>f^{(n+1)}(c) < 0</math>, then ''c'' is a local maximum.
* If ''(n+1)'' is even and <math>f^{(n+1)}(c) > 0</math>, then ''c'' is a local minimum.
* If ''(n+1)'' is [[parity (mathematics)|odd]] and <math>f^{(n+1)}(c) < 0</math>, then ''c'' is a strictly decreasing point of inflection.
* If ''(n+1)'' is odd and <math>f^{(n+1)}(c) > 0</math>, then ''c'' is a strictly increasing point of inflection.

Since ''(n+1)'' must be either odd or even, this analytical test classifies any stationary point of ''f'', so long as a nonzero derivative shows up eventually, where <math>f^{(n+1)}(c) \ne 0.</math> is the first non-zero derivative.