A function is continuous and the domain is all real numbers. (I have questions about critical points based on this type of function.)

Question

In class today we took notes on critical points. My teacher said that not every critical point would be a local max/min because exceptions can occur at discontinuities, and the same goes for global max/min. But I was wondering if I know the critical point is definitely a global max/min, then is that point is also a local max/min? Given that there is no discontinuity that provides the exception, every global max or min, is a local max or min right? or can the point only be labeled by one thing?

 

The second question I had about the notes is if a local max/min can occur where a function's derivative is undefined. This would mean the function has a horizontal tangent, so it would be a non differential point. Since this is a type of critical point then could it be a max/min?

 

I also didn't understand inflection points. Do inflection points occur when f''(x)=0? or at a local max/min? (I think it can occur at a max/min because the graph could change concavity, but I'm not sure.)

 

The last question I had was if we had to have a global max and min if x was restricted on a closed interval. (I don't think it would need to have one because couldn't the graph have the same minimum, so their is no definitive global min, and same thing for a max?

 

(Sorry, These notes today were just very confusing. The videos online aren't really helping me answer my questions.)

Answer 1

1.//But I was wondering if I know the critical point is definitely a global max/min, then is that point is also a local max/min? //

If the function is multi modal( many local min/maxi)  so far no algorithm can ensure it is global.

// Given that there is no discontinuity that provides the exception, every global max or min, is a local max or min right? or can the point only be labeled by one thing?//

 

Of course Global is also a local min/max. Global is simply the best among locals. First among equals!!!!!

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2.//The second question I had about the notes is if a local max/min can occur where a function's derivative is undefined.//

Derivative is zero and the edges where the derivative is undefined.But If we discuss about your case(domain is all real numbers) derivative is defined.For the condition of  continuity derive is to be defined at every point.

 

You have to rad about khun tucker conditions.