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 ƒ be a differentiable function on the interval I and let c be a point on it such that

1. $f'(c)=\cdots =f^{(n-1)}(c)=0;$
2. $f^{(n)}(c)\,$ exists and is non-zero.

Then,

1. if n is even
   1. $f^{(n)}(x)<0\implies x=c$ is a point of local maximum
   2. $f^{(n)}(x)>0\implies x=c$ is a point of local minimum
2. if n is odd
   1. $f^{(n)}(x)<0\implies x=c$ is a ghghgghhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhstrictly decreasing point of inflection
   2. $f^{(n)}(x)>0\implies x=c$ 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