Question

Let $f\left(x\right)=x{e}^{x(1-x)}$ then $f\left(x\right)$ is

• A. increasing on $[-1/2,1]$
• B. decreasing on $R$
• C. increasing on $R$
• D. decreasing on $[-1/2,1]$

Answer 1

The correct option is A increasing on $[-1/2,1]$

$f\left(x\right)=x{e}^{x(1-x)}$
$f^{\prime }\left(x\right)=e^{x(1-x)}+x(1-2x)e^{x(1-x)}$
$=(1+x-2x^2)e^{x(1-x)}$
$=-(x-1)(2x+1)e^{x(1-x)}$
Since $e^{x(1-x)}>0$ for all $x$
so $f^{\prime }\left(x\right)>0$ if and only if $(x-1)(2x+1)<0$
$\Rightarrow -\frac{1}{2}<x<1$
Thus $f$ increases on $[-1/2,1]$.
Ans: A