Question

Let $f(x)=x\left(e^{x^2}-e^{-x^2}\right)-2x-\int \left(e^{x^2}-e^{-x^2}\right)dx$. If $f(x)$ is decreasing in $\left(x_1,x_2\right)$ then $\left(x_1+x_2\right)$ equals

• A. 0
• B. $2e$
• C. $\frac{e^2-1}{e}$
• D. $\frac{3}{2}e$

Answer 1

Answer: (A)

$f(x)=x\left(e^{x^2}-e^{-x^2}\right)-2x-\int \left(e^{x^2}-e^{-x^2}\right)dx$

$f^{\prime }(x)=2x^2\left(e^{x^2}+e^{-x^2}\right)+\left(e^{x^2}-e^{-x^2}\right)-2-e^{x^2}+e^{-x^2}$
$f^{\prime }(x)=2x^2\left(e^{x^2}-e^{-x^2}\right)-2<0$
$\therefore e^{x^2}+e^{-x^2}<\frac{1}{x^2}$
$e^t+e^{-t}<\frac{1}{t};t>0$
decreasing in (-a, a)
$\therefore x_1+x_2=0$