Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Critical Points definition

  1. Nov 4, 2012 #1
    Are inflection points critical points?

    and what about at the value that f(x) undefined? Is that critical point too?
     
  2. jcsd
  3. Nov 4, 2012 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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, [itex]y= x^4[/itex] has [itex]y'= 4x^3[/itex] which is 0 only for x= 0 but the second derivative is [itex]y''= 12x^2[/itex] which is 0 at x= 0 but does NOT change sign there.

    On the other hand, [itex]y= x^3+ x[/itex] has [itex]y'= 3x^2+ 1[/itex] and [itex]y''= 6x[/itex]. 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.
     
  4. Nov 4, 2012 #3
    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
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Critical Points definition
  1. Critical point (Replies: 2)

  2. Critical points (Replies: 6)

  3. Critical points (Replies: 4)

  4. Critical Points (Replies: 2)

Loading...