Sum-frequency generation

2013-05-06T07:15:11Z

78.53.211.41:

{{unreferenced|date=September 2012}}
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 <math>f</math> be a real-valued, sufficient [[differentiable function]] on the interval <math>I \subset \R, \; c \in I</math> and <math>n \ge 1</math> an integer. If now holds
<math>f'(c)=\cdots=f^{(n)}(c)=0\quad \text{and}\quad f^{(n+1)}(c)\,\not= 0</math>

then, either

''n'' is odd and we have a local extremum at ''c''. More precisely:
#<math>f^{(n+1)}(c)<0 \Rightarrow c</math> is a point of a maximum
#<math>f^{(n+1)}(c)>0 \Rightarrow c</math> is a point of a minimum
or

''n'' is even and we have a (local) saddle point at ''c''. More precisely:
#<math>f^{(n+1)}(c)<0 \Rightarrow c</math> is a strictly decreasing point of inflection
#<math>f^{(n+1)}(c)>0 \Rightarrow c</math> is a strictly increasing point of inflection
.
This analytical test classifies any stationary point of <math>f</math>.

==See also==
*[[Extremum]]
*[[First derivative test]]
*[[Second derivative test]]
*[[Hessian_matrix#Second_derivative_test]]
*[[Saddle point]]
*[[Inflection point]]
*[[Stationary point]]

[[Category:Calculus]]

{{mathanalysis-stub}}

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Sum-frequency generation