Find the global maximum and the global minimum of the function $f$ on $\mathbb R$, where $f(x)= \frac {x^2-2x+4}{x^2+2x+4}, x \in \mathbb R$
My approach: I used the higher order derivative test, i.e. I calculated $f'(x), f''(x)$, found the points where $f'(x)=0$, and then showed that $f''(.)$ is greater or less than $0$ at those points. In this way, I found the LOCAL maximum and minimum.
Now, I know a result that if $f:I \rightarrow \mathbb R$ has a local maximum or minimum at a point $c \in I$ then $c$ is a global maximum on $N(c, \delta) \cap I$, where $\delta >0$
Is there any way to use this result to show that the local extremum I found here are global extremum on $\mathbb R$?
The method used in my book to solve this sum uses supremum and infimum, and I can't understand that working because the results are not given in the book. Could someone please give me references (pdfs or links) for global maximum or minimum involving supremum and infimum?