# Suppose that f(x) = ln(5 + x^2) Use interval notation to indicate where f(x) is concave up, concave down.

Question
Analyzing functions
Suppose that $$\displaystyle{f{{\left({x}\right)}}}={\ln{{\left({5}+{x}^{{2}}\right)}}}$$
Use interval notation to indicate where f(x) is concave up, concave down.

2020-11-09
$$\displaystyle{f{{\left({x}\right)}}}={\ln{{\left({5}+{x}^{{2}}\right)}}}$$
$$\displaystyle{f}{1}{\left({x}\right)}=\frac{{{2}{x}}}{{{5}+{x}^{{2}}}}$$
$$\displaystyle{f}{2}{\left({x}\right)}=\frac{{{2}{\left({5}–{x}^{{2}}\right)}}}{{\left({5}+{2}^{{2}}\right)}^{{2}}}={0}$$
$$\displaystyle{x}=\sqrt{{5}},-\sqrt{{5}}$$
Downward $$\displaystyle{\left(-\infty,-\sqrt{{5}}\right)}\cup{\left(\sqrt{{5}},\infty\right)}$$
Upwards $$\displaystyle{\left(-\sqrt{,}\sqrt{{5}}\right)}$$

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$$\displaystyle{f{{\left({x}\right)}}}={8}{x}^{{2}}+{8}{x}+{4}.$$