Question

What is different between critical point and inflection point?

Answer 1

There seem to be two definitions of "critical point" in use. But with that in mind:

Explanation:

A critical number for function `f`, is a number in the domain of `f` where `f'(x) = 0` or `f'(x)` does not exist.
These are the numbers at which Fermat's Theorem on local extrema tells us `f` could have a local maximum or local minimum value.

In some usages "critical point" is synonymous with this "critical number".
In other usages a critical point is a point `(x, f(x))` with `f'(x)=0` or `f'(x)` does not exist.

An inflection point for the graph of function `f` is a point on the graph at which the concavity of the graph changes.
For twice differentiable functions, this is a point on the graph of `f` at which `f''(x)` changes sign.

Students sometimes use "inflection point" to mean an `x` value at which the concavity changes.

The difference is illustrated by `f(x) = 1/x` which is concave down on `(-oo, 0)` and concave up on `(0, oo)`.
In the "point" usage, the graph has no inflection point, because there is no point on the graph where concavity changes.