Higher-order derivative test

In mathematics, the higher-order derivative test is used to find maxima, minima and points of inflection in an n^th degree polynomial's curve.

The test

Let $f$ be a differentiable function on the interval $I$ and let $c$ be a point on it such that

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

such that n is made as large as possible.

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 $\implies x=c$ is a point of inflection

See also

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