supremum and infimum

[doubt]

hello to everyone
i have some sort of problem to find infimum and supremum of funtion
as a difination i know them well but really i can t apply it
someone can explain me step by step what should i do in order to find supremum of this simple function x+1/x^2+x+3
guys i have really hard test please help me to understand this hard concept
if there is some sort o general way to do with all function or at least the important things to know
i will be glad to know it thank you so much

 

[Deleted member 4993]

hello to everyone
i have some sort of problem to find infimum and supremum of function
as a definition i know them well but really i can t apply it
someone can explain me step by step what should i do in order to find supremum of this simple function x+1/x^2+x+3
guys i have really hard test please help me to understand this hard concept
if there is some sort o general way to do with all function or at least the important things to know
i will be glad to know it thank you so much

Is your function:

$\displaystyle f(x) \ = \ x \ + \ \frac{1}{x^2} \ + \ x \ + \ 3$

If not - please post it with correct grouping symbols.

What are your thoughts?

Please share your work with us ...even if you know it is wrong

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/th...Before-Posting

 

[doubt]

Is your function:

$\displaystyle f(x) \ = \ x \ + \ \frac{1}{x^2} \ + \ x \ + \ 3$

If not - please post it with correct grouping symbols.

What are your thoughts?

Please share your work with us ...even if you know it is wrong

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/th...Before-Posting

(x+1)/(x^2+x+3)

Sorry about that I asked so many people to help me in this problem I know definition well
I will write everything that I know so far
but I show my work just for this function maybe I can go further but the problem is what should I do when a professor says ok find the supremum of another function or infimum of any function
for the fact I know supremum is least upper bound and infimum is greatest lower bound
and an upper bound is the element of the function that greatest or equal to all of them
and the lower bound is the element that smaller or equal to all the element of the set
first of all, I have to know how a function behaves so I set denominator equal = 0 to find a domain
and maybe vertical asymptote then I look for horizontal asymptote if there is any
I set x^2+x+3 = 0 but there is no solution so there is no hole and vertical asymptote
then I take a limit of the function as x approach +-infinity so that is zero
then I know x-y intercepts are -1 and 1/3
So I have a vision of the graph
but now what should I do should I say limit as x approach 1/3 from the right and then 0 from the left
when I did that I saw something strange as x approaches 1/3 function goes to 3 so what is that number

I find derivative of that function
-\dfrac{x^2+2x-2}{\left(x^2+x+3\right)^2}
and then if I set it equal zero gives me these point from denominator should I worry about denominator or just nominator

x=√3−1≈0.7320508075688773​

x=−√3−1≈−2.732050807568877


​

​

and then I have some point for nominator
but after all of this point how I can I know If I have supremum or infimum
thank you so much

 

[Ishuda]

(x+1)/(x^2+x+3)

Sorry about that I asked so many people to help me in this problem I know definition well
I will write everything that I know so far
but I show my work just for this function maybe I can go further but the problem is what should I do when a professor says ok find the supremum of another function or infimum of any function
for the fact I know supremum is least upper bound and infimum is greatest lower bound
and an upper bound is the element of the function that greatest or equal to all of them
and the lower bound is the element that smaller or equal to all the element of the set
first of all, I have to know how a function behaves so I set denominator equal = 0 to find a domain
and maybe vertical asymptote then I look for horizontal asymptote if there is any
I set x^2+x+3 = 0 but there is no solution so there is no hole and vertical asymptote
then I take a limit of the function as x approach +-infinity so that is zero
then I know x-y intercepts are -1 and 1/3
So I have a vision of the graph
but now what should I do should I say limit as x approach 1/3 from the right and then 0 from the left
when I did that I saw something strange as x approaches 1/3 function goes to 3 so what is that number

I find derivative of that function
$\displaystyle -\dfrac{x^2+2x-2}{\left(x^2+x+3\right)^2}$
and then if I set it equal zero gives me these point from denominator should I worry about denominator or just nominator

x=√3−1≈0.7320508075688773​

x=−√3−1≈−2.732050807568877


​

​

and then I have some point for nominator
but after all of this point how I can I know If I have supremum or infimum
thank you so much

Thought I would add my 2 cents. First, to use LaTex on the formum enclose the equations in the [ tex ][ /tex ] tag pair without the spaces. I fixed the one for you above in this post.

To get to your problem: For this problem you are almost there: Evaluate the function at the two points since you will likely need those You have several points for the graph of the function, ($\displaystyle -\infty$, 0-), (-2.73, -0.22), (-1, 0), (0, 0.33), (0.73, 0.41), ($\displaystyle \infty$, 0+). Since there are no horizontal nor vertical asymptotes nor holes, you should be able to sketch the graph now and see that the maximum (and thus the supremum) of the function is at (0.73, 0.41) and the infimum is at (-2.73, -0.22).

That is, if a function takes on its absolute maximum/minimum value then that value is the supremum/infimum of the function.


To determine those possible maximum/minimum values for continuous differential functions you could, for example, read about it here
https://en.wikipedia.org/wiki/Derivative_test#First_derivative_test
However, you generally you only need the first two derivatives to get: If f'=0 and f''>0 there is a (possibly relative) minimum, i.e. f(x)=x² with a relative (and absolute) minimum at f(0); if f'=0 and f''<0 there is a (possibly relative) maximum, i.e. f(x)=-x² with a relative (and absolute) maximum at f(0).

Just to sort of complete things, let's restrict the domain under consideration to $\displaystyle (-\infty,\, -2.73]$ and again ask for the supremum/infimum. Well the absolute minimum of the function is again (-2.73, -0.22), and thus that is its infimum but what about the supremum. For this problem we can see the 0 is the least upper bound and thus is the supremum of the function. We thus have that if the limit of the function as x approaches x₀ is y₀ and y₀ is the least upper bound/ greatest lower bound of the function then y₀ is the supremum/infimum of the function.