SHOW YOUR REASONING QUESTION $($ALL STEPS$)$ FOLLOWS

$(10$ points$)$ Consider the following function and its first two derivatives:

$y=\frac{1}{1+x^2}$;$,y^{\text{'}}=-\frac{2x}{(1+x^2{)}^2}$;$,y^{\text{'}\text{'}}=\frac{2(3x^2-1)}{x^2+1}$

Find all local extrema, points of inflection, horizontal and vertical asymptotes $($if any$)$

and give a calculus sketch of this function from $-5$ to $5.$ On the sketch, mark extreme

points, points of inflection, and for any asymptotes, use dashed lines.