# Non-Degenerate Extrema

1. Sep 1, 2010

### 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.)

2. Sep 1, 2010

### Office_Shredder

Staff Emeritus
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

3. Sep 7, 2010

### 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?

4. Sep 7, 2010

### Office_Shredder

Staff Emeritus
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 xi converging to our critical point x0 such that f'(xi)=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=xi 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 xi such that f'(xi) does not exist converging to x0, which means that f''(x0) can't exist either

5. Sep 7, 2010

### Apteronotus

I'm grateful. Thank you.