Are Non-Degenerate Extrema Isolated in Functions of One Variable?

[Apteronotus]

If it is said that a function has nondegenerate extrema does this simply mean that the extrema are isolated?

(The function in question is of one variable.)

Thanks in advance

 

[Office_Shredder]

A nondegenerate critical point is one in which the hessian matrix (the matrix of partial second derivatives) is nonsingular. A nondegenerate extremum then is just a nondegenerate critical point which is an extremum

 

[Apteronotus]

Office_Shredder
thank you for your reply. Before posting I searched online for the definition and came across the definition you posted.

Hence let me rephrase my question:
What is the geometrical significance of a non-singular Hessian matrix (in the context stated above)?
-- Is an extremum point whose Hessian is non-singular an isolated extrema?

 

[Office_Shredder]

Looking back that was a dumb reply since you said that your function only has one variable.

The critical point is nondegenerate if the second derivative at the critical point is non-zero. There are certainly degenerate critical points that are isolated: for example the point 0 for the function x⁴. If your definition of critical point allows for points where the derivative does not exist, then 0 for the function |x| also counts.

If a critical point is nondegenerate, say it's called x₀, then it has to be isolated. If there is a sequence of points x_i converging to our critical point x₀ such that f'(x_i)=0 for all i, then if the second derivative exists, it must be zero (in the difference quotient, looking at just choices of h such that x+h=x_i shows that). If we're in the case where derivatives don't exist for the other critical points, but the derivative of x₀ does exist, then the second derivative can't exist because we have a sequence of points x_i such that f'(x_i) does not exist converging to x₀, which means that f''(x₀) can't exist either

 

[Apteronotus]

I'm grateful. Thank you.