Consider the function f(x)=2x+2x1\displaystyle {f{{\left({x}\right)}}}={2}{x}+{2}{x}^{{-{1}}}. For this function there are four important intervals: (,A]\displaystyle {\left(-\infty,{A}\right]}, [A,B)\displaystyle {\left[{A},{B}\right)},(B,C]\displaystyle {\left({B},{C}\right]}, and [C,)\displaystyle {\left[{C},\infty\right)} where A\displaystyle {A}, and C\displaystyle {C} are the critical numbers and the function is not defined at B\displaystyle {B}.
Find A\displaystyle {A}  
and B\displaystyle {B}  
and C\displaystyle {C}  

For each of the following intervals, tell whether f(x)\displaystyle {f{{\left({x}\right)}}} is increasing (type in INC) or decreasing (type in DEC).
(,A]\displaystyle {\left(-\infty,{A}\right]}:
[A,B)\displaystyle {\left[{A},{B}\right)}:
(B,C]\displaystyle {\left({B},{C}\right]}:
[C,)\displaystyle {\left[{C},\infty\right)}

Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f(x)\displaystyle {f{{\left({x}\right)}}} is concave up (type in CU) or concave down (type in CD).
(,B)\displaystyle {\left(-\infty,{B}\right)}:
(B,)\displaystyle {\left({B},\infty\right)}: