Higher Order Derivatives

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

$$\lim_{x\to a}{\frac{f^{(n-1)}(x)-f^{(n-1)}(a)}{x-a}} = L \in \mathbb{R} = f^{(n)}(a)$$

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 $a$.

Note

$f \text{ is } n \text{-th differentiable at } a \implies f^{(n-1)} \text{ is continuous at } a$

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 $n=1$: $$

$$f(x)=f(c)+f'(c)(x-c)+\frac{f''(\zeta)}{2!}(x-c)^2$$ $$f(x)-\text{Tangent line}=\frac{f''(\zeta)}{2!}(x-c)^2$$