I was recently reading about classifying the extrema of a continuous function of a single variable. I came across the information that if the second derivative is zero then we can examine the higher derivatives such as $f^{\prime\prime\prime}$,$f^{(4)}$ and so on. Suppose $f^{(n)}$ is the first non-zero derivative. If n is odd, then the point is an inflection point and if n is even then a positive nth derivative means a minimum and a negative nth derivative means a maximum.
Now, my question is, can anyone come up with an example of a function $f(x)$ where this information about the nth derivative can be applied? I wanted to add one to my notes. Further, can anyone come up with an example of both a inflection point and a maximum or minimum?