Question

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

• A. increasing on $[\frac{-1}{2},1]$
• B. decreasing on R
• C. increasing on R
• D. decreasing on $[\frac{-1}{2},1]$

Answer 1

The correct option is A increasing on $[\frac{-1}{2},1]$

$f\left(x\right)=x{e}^{x(1-x)}$
$\Rightarrow f^{\prime (}x)=e^{x(1-x)}+(1-2x)x{e}^{x(1-x)}$
$=-e^{x(1-x)}(2x^2-x-1)=-e^{x(1-x)}(2x+1)(x-1)$

$\therefore f\left(x\right)$ is increasing on $[\frac{-1}{2},1]$