Mathematics - Application of Derivative Question with Solution | TestHub

$f(x)=\frac{e^x}{1+x^2},g(x)=f^{\prime }(x)$ $g^{\prime }(x)=f^{\prime \prime }(x)=\frac{e^x(x-1)}{1+x^2}\left(x^3-3x^2+5x+1\right)$ Let $\mathrm{h}(\mathrm{x})={\mathrm{x}}^3-3{\mathrm{x}}^2+5\mathrm{x}+1,{\mathrm{h}}^{\prime }(\mathrm{x})=3{\mathrm{x}}^2-6\mathrm{x}+5,\mathrm{D}<0$ So, $\mathrm{h}(\mathrm{x})$ is strictly increasing will exactly have one real root. Product of roots < 0, so the real root is negative. So, ${\mathrm{g}}^{\prime }(\mathrm{x})$ will change sign only at two points.