Higher Order Derivatives

Suppose $f$ is a function defined on $(a,b)$. $f$ is $n$ times differentiable or $n$-th differentiable iff:

Here $f^{(n)}$ denotes $n$-th derivative of $f$. And $f^{(0)}$ means the function itself.

$f^{(n)}(a)$ is the $n$-th derivative of $f$ at $a$.

Second Derivative Test

Suppose $f'(x)=0$ and $f''(x)$ is continuous at $c$:

• $f''(c) \gt 0 \implies$ a local minimum is at $c$. Converse is not true.
• $f''(c) \lt 0 \implies$ a local maximum is at $c$. Converse is not true.

The above conclusion is from Taylor’s theorem when $n=1$: