Question on Stationary Points/Inflexion Points... (2 Viewers)

[gr_111]

Is there ever any point on a graph where f'(x) = 0 and f"(x) = 0 and it is not a horizontal inflexion point?

In other words, is f'(x)=0, f"(x)=0 always a horizontal inflexion??

Thanks

 

[Carrotsticks]

Is there ever any point on a graph where f'(x) = 0 and f"(x) = 0 and it is not a horizontal inflexion point?

In other words, is f'(x)=0, f"(x)=0 always a horizontal inflexion??

Thanks

No, not always a horizontal inflexion.

Counterexample: y=x^4.

 

[gr_111]

Counterexample: y=x^4.

Ok... didn't think about that one. Thanks.

 

[Fus Ro Dah]

Ok... didn't think about that one. Thanks.

To really confirm for some classes of functions, say Polynomials, you could actually differentiate the function until it reduces to a constant and by observing whether that constant is positive, negative, or equal to 0, we can immediately observe its 'nature'.

We have 3 cases essentially, quite similarly to how we have 3 cases for the discriminant being less than 0, equal to 0, or greater than 0.

Case #1

Suppose we have some function f satisfying the following:



and



then the point is a minimum but not necessarily

Case #2

Suppose we have some function f satisfying the following:



and



then the point is an inflexion point.

Case #3

Suppose we have some function f satisfying the following:



and



then the point is a minimum but not necessarily