Question

I am having trouble with this definition of critical point, wikipedia doesn't mention that critical points occur where the derivative is undefined
http://en.wikipedia.org/wiki/Critical_point_%28mathematics%29

Answer 1

In my book it says
a critical number/point of a function is a number c in the domain of f such that either f'(c) = 0 or f'(c) does not exist (undefined)

Answer 2

here is a nice set of notes on the topic

http://tutorial.math.lamar.edu/Classes/CalcI/CriticalPoints.aspx

Answer 3

something is wrong with wiki's definition of critical point?

Answer 4

this is wiki's definition:
"A critical point or stationary point of a differentiable function of a single real variable, f(x), is a value x0 in the domain of f where its derivative is 0: f′(x0) = 0."

Then later wiki says in the same example, and i dont understand how it falls out of the definition:
"The function f(x) = x^(2/3) is defined for all x and differentiable for x ≠ 0, with the derivative f′(x) = 2/3x^(−1/3). Since f′(x) ≠ 0 for x ≠ 0, the only critical point of f is x = 0. The graph of the function f has a cusp at this point with vertical tangent. The corresponding critical value is f(0) = 0."

Answer 5

I wasn't sure so I googled.
Every source I've checked so far DOES include a cusp as a critical point.

So yes maybe wiki is a bit inaccurate on this.

Answer 6

or even a function like y = |x| is not differentiable at x = 0, and this is a critical point.

Answer 7

should i edit wiki?

Answer 8

Haha I have no idea, I've never tried that XD

Answer 9

it seems like glaring omission

Answer 10

wiki is usually exhaustive on any given topic :/

Answer 11

i think there should be a distinction made between 'stationary point' , where the derivative is defined, and 'critical point' where the derivative may or may not be defined.

Answer 12

Yah I like the distinction.
I mean if the function isn't differentiable there, it's not really a stationary point.
It's just something else and apparently we still call it a critical point. hmm

Answer 13

here compare the following:
http://mathworld.wolfram.com/CriticalPoint.html
http://mathworld.wolfram.com/StationaryPoint.html

Answer 14

Ah nice, wolfram's definition seems a bit clearer :)

Answer 15

im not quite sure how to edit wikipedia, let me try

Answer 16

Careful! Don't break the interwebs! :O

Answer 17

i cant edit the first paragraph

Answer 18

maybe it is satisfied vacuously?

Answer 19

"Critical points can occur for values of \(x\) where \(f'(x)=0\) or is undefined," is how I learned it.

Answer 20

yes me too :)

Answer 21

Critical points are where

f'(x)=0

f''(x)=0 that simple.

Answer 22

Idk about wiki. I think their info on most things may be accurate. But they suck at explaining.

Answer 23

critical points , is the points which the curve change it direction on it :")