Chapter 4.5, Problem 79E

To determine

(a)

To show:

The equation $\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{{\left(x-c\right)}^n}=\frac{1}{n!}{f}^{\left(n\right)}\left(c\right)$ using L’ Hôpital’s rule $n$ times.

To determine

(b)

To show:

That if $n$ is even then $f\left(c\right)$ is a local minimum if $f^{\left(n\right)}\left(c\right)>0$ and $f\left(c\right)$ is a local maximum if $f^{\left(n\right)}\left(c\right)<0$.

To determine

(c)

To show:

That if $n$ is odd then $f\left(c\right)$ is neither a local minimum nor a local maximum.