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

Determine all critical values of $f$. If there is more than one, enter the values as a comma-separated list.
Critical value(s) =

Construct a first derivative sign chart for $f$ and thus determine all intervals on which $f$ is increasing or decreasing. If there is more than one, enter the intervals as a comma-separated list. Use interval notation: for example, (-17,20) is the interval $-17 < x < 20$, and (-inf, 40) is the interval $x<40$.
Interval(s) where $f$ is increasing:
Interval(s) where $f$ is decreasing:

Does $f$ have a global maximum? If so, enter its value. If not, enter DNE.
Global maximum =

Determine the following limits.
$\displaystyle{\lim_{x\rightarrow \infty}} f(x) =$
$\displaystyle{\lim_{x\rightarrow -\infty}} f(x) =$

Explain why $f(x) > 2$ for every value of $x$.

Does $f$ have a global minimum? If so, enter its value. If not, enter DNE.
Global minimum =

In order to get credit for this problem all answers must be correct.