Increasing-Decreasing & Concavity on Intervals
With this applet you control whether we can see the local extrema or points of inflection. For each, the appropriate points are shown as well as vertical lines passing through each point.
If you view the local extrema, you can look in between the vertical lines and see that the function is either always increasing or always decreasing on each resulting interval.
If you view the points of inflection, you can look in between the vertical lines and see that the function is either always concave up or always concave down on each resulting interval.