Let f(x) = (x^2+1) / (x^2-2).
The domain of f is (-oo,-sqrt2) uu (-sqrt2, sqrt2) uu (sqrt2,oo).
To find critical numbers, we find f'(x) = (-6x)/(x^2-2)^2 which is 0 at x=0 and is never undefined on the domain of f.
The only critical number is 0.
To investigate concavity and inflection points, we find f''(x) = (6(3x^2+2))/(x^2-2)^3 which is never 0 and is undefined at -sqrt2 and at sqrt2
Sign of f''
{: (bb "Interval", bb"Sign of "f'',bb" Concavity"),
((-oo,-sqrt2)," "" " +" ", " "" Up"),
((-sqrt2,sqrt2), " "" " -, " " " Dn"),
((sqrt2 ,oo), " "" " +, " "" Up")
:}
The graph of f is concave up on the intervals (-oo,-sqrt2) and (sqrt2,oo). The graph is concave down on (-sqrt2,sqrt2).
Although the concavioty changes at x=+-sqrt2, there is no point on the graph at x=+-sqrt2, so there are no inflection points.
