I find the lack of information on the internet about the third derivative astounding.
When the first derivative of a function touches or crosses the x-axis, it is a critical point and possibly an extrema. If $f'(x)\gt0$, the slope of $f$ is increasing; if $f'(x)<0$, the slope of $f$ is decreasing.
When the second derivative of a function crosses the x-axis there is an inflection point, and the concavity reverses. If $f''(x)>0$, $f$ is concave up; if $f''(x)<0$, $f$ is concave down.
What visual effects does the third derivative have on its function? Do higher order derivatives have any clearly visible effects?
