Question

If $f(x)=x{e}^{x(1-x)}$, 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)

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