Study of ln(x squared plus 4)

Text

Study the function $y=\ln(x^{2}+4)$: domain, symmetry, minimum and inflections.

Solution

Domain $\mathbb{R}$; even function. First derivative: $y'=\dfrac{2x}{x^{2}+4}=0\Rightarrow x=0$: minimum $(0,\ln4)\approx(0,1.386)$. Second derivative: $y''=\dfrac{8-2x^{2}}{(x^{2}+4)^{2}}=0\Rightarrow x=\pm2$: two inflections. As $x\to\pm\infty$, $y\to+\infty$. $\boxed{\text{min }(0,\ln4)\approx(0,1.386);\ \text{inflections at }x=\pm2}$