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