Answer
$$\eqalign{
& {\text{Domain }}\left( { - \infty ,0} \right) \cup \left( {0,\infty } \right) \cr
& {\text{Relative maximum }}\left( {1.0962,9.0457} \right) \cr
& {\text{Relative minimum }}\left( { - 1.0962, - 9.0457} \right) \cr
& {\text{No Inflection points}} \cr
& {\text{Vertical Asymptote }}x = 0 \cr
& {\text{Horizontal Asymptote }}y = 0 \cr} $$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \frac{{20x}}{{{x^2} + 1}} - \frac{1}{x} \cr
& {\text{Using a CAS }}\left( {{\text{WolframAlpha Website}}} \right){\text{ we obtain:}} \cr
& {\text{Domain: }}\left\{ {\left. {x \in R} \right|x \ne 0} \right\} \cr
& {\text{Odd Function }}\left( {{\text{Symmetry with respect to origin}}} \right) \cr
& {\text{Max at }}x = \sqrt {\frac{1}{{19}}\left( {11 + 2\sqrt {35} } \right)} \cr
& {\text{Min at }}x = - \sqrt {\frac{1}{{19}}\left( {11 + 2\sqrt {35} } \right)} \cr
& {\text{Inflection points at:}} \cr
& x = - \sqrt {\frac{{11}}{{19}} + \frac{{2\sqrt {35} }}{{19}}} \cr
& x = \sqrt {\frac{{11}}{{19}} + \frac{{2\sqrt {35} }}{{19}}} \cr
& {\text{Vertical Asymptote }}x = 0 \cr
& {\text{Horizontal Asymptote }}y = 0 \cr
& \cr
& {\text{Graph}} \cr} $$
