Extreme points

[Cata]

Hello. Can somebody tell me in what cases a point x0 is not an extreme point, even tho f'(x0)=0? Thank you in advance!

 

[Dr.Peterson]

Hello. Can somebody tell me in what cases a point x0 is not an extreme point, even tho f'(x0)=0? Thank you in advance!

What tests have you been taught to do, to determine whether a critical point is an extreme point? Is there something involving the second derivative, for example? And have you seen any examples where it turns out not to be an extreme point?

 

[JeffM]

Hello. Can somebody tell me in what cases a point x0 is not an extreme point, even tho f'(x0)=0? Thank you in advance!

Consider as an example the function

\(\displaystyle f(x) = x^3 \implies f'(x) = 3x^2 \implies f'(x) > 0 \text { unless } x = 0.\)

Note that

\(\displaystyle f'(0) = 0 \text { but } x < 0 \implies f(x) < 0 = f(0) \text { and } x > 0 \implies f(x) > 0 = f(0).\)

Thus, there is no extreme value of f(x) when x = 0.

For the first derivative to indicate an extremum, the derivative must have opposite signs immediately to the right and left of the extreme point. Thus, a first derivative of zero does not guarantee an extremum.

 

[maxwell]

Hello. Can somebody tell me in what cases a point x0 is not an extreme point, even tho f'(x0)=0? Thank you in advance!

A simple example: Let C be a constant real number and let f(x)=C for each real number x. Then, for each x, f '(x)=0 and f(x)=C and is therefore not an extreme point.