Are All Inflection Points Also Critical Points?

[daivinhtran]

Are inflection points critical points?

and what about at the value that f(x) undefined? Is that critical point too?

 

[HallsofIvy]

A "critical point" is a point at which the derivative either does not exist or is 0. An "inflection point" is where the second derivative changes sign. Obviously, that can only happen where the second derivative is 0 but the other way is not true. For example, y= x^4 has y'= 4x^3 which is 0 only for x= 0 but the second derivative is y''= 12x^2 which is 0 at x= 0 but does NOT change sign there.

On the other hand, y= x^3+ x has y'= 3x^2+ 1 and y''= 6x. At x= 0, y'' changes sign but the first derivative is not 0 there. x= 0 is an inflection point but is NOT critical point.

For you last question, no. f(x) must be defined at a critical point.

 

[Bipolarity]

Are inflection points critical points?

and what about at the value that f(x) undefined? Is that critical point too?

To be a critical point:
- It must be in the domain of the function, meaning the function must be defined at the point.
- The first derivative must either be 0 or undefined.

Some inflection points are also critical points, but not all inflection points are critical.

BiP