Consider the function $f$ defined by

$f(x)=\frac{(x-1{)}^2}{x^2+1},$xinR.

$(1$ mark$)$

Find the maximal domain of $f,$ the intercepts of its graph $y=f(x)$ with the $y-$axis and $x-$axis, and its sign $($the regions where $f$ is positive or negative$).$

$2.(1$ mark$)$

Compute the limits of $f$ at the boundaries of its domain and decide whether $f$ has asymptotes of any type.

$3.(1$ mark$)$

Compute $f^{\text{'}},$ identify the stationary points for $f,$ find its maxima and minima, as well as the regions where $f$ is increasing and decreasing.

$4.(1$ mark$)$

Compute $f^{\text{'}\text{'}},$ identify the flexes with non$-$horizontal tangent specifying their type, and find the regions where $f$ is convex and is concave.

$5.(1$ mark$)$

Finally, sketch the graph of $f$ including all the above information.