Question

If $f( x )= xe ^{ x (1- x )}$, then $f ( x )$ is

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

Answer 1

Answer: (A)

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