\(\displaystyle{f{{\left({x}\right)}}}={\ln{{\left({5}+{x}^{{2}}\right)}}}\)

\(\displaystyle{f}{1}{\left({x}\right)}=\frac{{{2}{x}}}{{{5}+{x}^{{2}}}}\)

\(\displaystyle{f}{2}{\left({x}\right)}=\frac{{{2}{\left({5}–{x}^{{2}}\right)}}}{{\left({5}+{2}^{{2}}\right)}^{{2}}}={0}\)

\(\displaystyle{x}=\sqrt{{5}},-\sqrt{{5}}\)

Downward \(\displaystyle{\left(-\infty,-\sqrt{{5}}\right)}\cup{\left(\sqrt{{5}},\infty\right)}\)

Upwards \(\displaystyle{\left(-\sqrt{,}\sqrt{{5}}\right)}\)

\(\displaystyle{f}{1}{\left({x}\right)}=\frac{{{2}{x}}}{{{5}+{x}^{{2}}}}\)

\(\displaystyle{f}{2}{\left({x}\right)}=\frac{{{2}{\left({5}–{x}^{{2}}\right)}}}{{\left({5}+{2}^{{2}}\right)}^{{2}}}={0}\)

\(\displaystyle{x}=\sqrt{{5}},-\sqrt{{5}}\)

Downward \(\displaystyle{\left(-\infty,-\sqrt{{5}}\right)}\cup{\left(\sqrt{{5}},\infty\right)}\)

Upwards \(\displaystyle{\left(-\sqrt{,}\sqrt{{5}}\right)}\)