Infimum vs Absolute minimum

[mdrak12]

I was beginning to understand the concept of infimums and supremums of bounded sets. However now I'm confused and need clarification. I have the polynomial function g(x)=x²+2x-3. Now to go about finding the infimum am I looking at the absolute minimum or should I look at the roots of the graph?

 

[Steven G]

I was beginning to understand the concept of infimums and supremums of bounded sets. However now I'm confused and need clarification. I have the polynomial function g(x)=x²+2x-3. Now to go about finding the infimum am I looking at the absolute minimum or should I look at the roots of the graph?

It depends (not really!) on how you define infimums. Can you please state the definition and try to use it for your function? Please post back with your work.

 

[mdrak12]

A number t is a greatest lower bound of B if:

(i) t is a lower bound for the set B, if every element in B is greater than or equal to t
(ii) t is the greatest lower bound to the set B. That is if t is a lower bound for B and there is no other lower bound (b) for B that is greater than (b)

My first intuition is to use the absolute minimum (-1,-4). Thus g(x) is bounded below by -4 and reaches its bound at x=-1. I'll need to work on how to formally prove this, but is this the right track?

 

[Steven G]

Yes, you are on the right track.
So first show that -4 is a LB and then show that it is the GLB.
Then report back. If you need any hints then just ask but really try first!

 

[Harry_the_cat]

Are we talking here about local minimums and global minimums? The term "infimum" is new to me.

 

[Otis]

... The term "infimum" is new to me.

Try this.

 

[Harry_the_cat]

Thanks Otis. I should have done that for myself!!