What is different between critical point and inflection point?
There seem to be two definitions of "critical point" in use. But with that in mind:
A critical number for function
These are the numbers at which Fermat's Theorem on local extrema tells us
In some usages "critical point" is synonymous with this "critical number".
In other usages a critical point is a point
An inflection point for the graph of function
For twice differentiable functions, this is a point on the graph of
Students sometimes use "inflection point" to mean an
The difference is illustrated by
In the "point" usage, the graph has no inflection point, because there is no point on the graph where concavity changes.