2.4.21 Graphing functions using derivatives

condition: Given a function \( f \) : \( f(x)=x \tan ^{-1}(x) \). Find the domain of the function \(f\). a. Prove that \( f^{\prime \prime}(x)=\frac{2}{\left(1+x^{2}\right)^{2}} \) b. Find where the graph of the function \( f \) bends up and down, find all the singular and critical points for the function \( f \), and find the possible absolute maximum and minimum of the function \( f \). c. Find all asymptotes of the function \(f\). d. Plot the graph of the function \(f\).