Studio di funzioni – Esercizio 67

\[ f”\left(x\right)=\left(x^{2}+1\right)^{-\frac{3}{2}}+\left(x+1\right)\left(-\frac{3}{2}\right)\left(x^{2}+1\right)^{-\frac{5}{2}}\cdot2x \] \[ f”\left(x\right)=\left(x^{2}+1\right)^{-\frac{3}{2}}-3x\left(x+1\right)\left(x^{2}+1\right)^{-\frac{5}{2}} \] \[ f”\left(x\right)=\frac{1}{\sqrt{\left(x^{2}+1\right)^{3}}}-\frac{3x\left(x+1\right)}{\sqrt{\left(x^{2}+1\right)^{5}}} \] \[ f”\left(x\right)=\frac{x^{2}+1-3x^{2}-3x}{\sqrt{\left(x^{2}+1\right)^{5}}} \] \[ f”\left(x\right)=\frac{-2x^{2}-3x+1}{\sqrt{\left(x^{2}+1\right)^{5}}} \] \[ f”\left(x\right)\geq0\rightarrow2x^{2}+3x-1\leq0 \] \[ f”\left(x\right)\geq0\rightarrow x\geq\frac{-3-\sqrt{17}}{4}\:\wedge\:x\leq\frac{-3+\sqrt{17}}{4} \] e otteniamo due flessi: \[ x_{F1}=\frac{-3-\sqrt{17}}{4} \] \[ x_{F2}=\frac{-3+\sqrt{17}}{4} \]