Question

Let $f(x)=\frac{e^x}{1+x^2}$ and $g(x)=f^{\prime }(x)$ then

• A. $g(x)$ has two local maximum and two local minimum points
• B. $g(x)$ has exactly one local maximum and one local minimum point
• C. $x=1$ is a point of local maximum for $g(x)$
• D. There is a point of local maximum for $g(x)$ in interval $(-1,0)$

Answer 1

Answer: (B), (D)

$\Theta g(x)=f^{\prime }(x)=\frac{\left(1+x^2\right)e^x-2x{e}^x}{{\left(1+x^2\right)}^2}=\frac{(x-1{)}^2{e}^x}{{\left(1+x^2\right)}^2}$
$g^{\prime }(x)=\frac{(x-1)\left(x^3-3x^2+5x+1\right)e^x}{{\left(x^2+1\right)}^3}$
here $x^3-3x^2+5x+1$ is strictly increasing and has a root in $(-1,0)$.