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

• Correct answer
  A

  increasing on $\left[-\frac{1}{2}, 1\right]$
• B

  decreasing on $R$
• C

  increasing on $R$
• D

  decreasing on $\left[-\frac{1}{2}, 1\right]$

Solution

We have, $f(x)=x e^{x(1-x)}$ On differentiating both sides w.r.t. $x$, we get $\begin{aligned} & f^{\prime}(x)=e^{x(1-x)}+x(1-2 x) e^{x(1-x)} \\ & \Rightarrow \quad f^{\prime}(x)=\left(1+x-2 x^2\right) e^{x(1-x)} \\ & \Rightarrow \quad f^{\prime}(x)=-(x-1)(2 x+1) e^{x(1-x)} \end{aligned}$ Since, $e^{x(1-x)}\gt0$ for all $x$. Therefore, signs of $f^{\prime}(x)$ for different values of $x$ are as shown below:
Clearly, $f(x)$ is increasing on $\left[-\frac{1}{2}, 1\right]$ and decreasing on $\left(-\infty,-\frac{1}{2}\right] \cup[1, \infty)$.

Hence, the correct answer is option: 'A'.