Higher Derivative Test

Higher Derivative Test

Suppose:

$$f'(c)=f''(c)=\cdots=f^{(n-1)}(c)=0$$

but:

$$f^{(n)}(c)\neq0$$

If $$n$$ is even and:

$$f^{(n)}(c)>0$$

then $$f(c)$$ is a local minimum.

If $$n$$ is even and:

$$f^{(n)}(c)<0$$

then $$f(c)$$ is a local maximum.

If $$n$$ is odd, $$x=c$$ is not a local maximum or minimum.

Usage

- Used when several consecutive derivatives vanish at a stationary point.