Mathematics - Application of Derivative Question with Solution | TestHub

$$\begin{array}{rl}& f(x)=x{e}^{x(1-x)},x\in R\\ & f^{\prime }(x)=e^{x(1-x)}\cdot \left[1+x-2x^2\right]\end{array}$$ $$\begin{array}{rl}& =-e^{x(1-x}\cdot \left[2x^2-x-1\right]\\ & =-2e^{x(1-x}\cdot \left[\left(x+\frac{1}{2}\right)(x-1]\right)\\ & f^{\prime }(x)=-2e^{x(1-x)}A\\ & \text{ where }A=\left(x+\frac{1}{2}\right)(x-1)\end{array}$$ Now, exponential function is always $+$ ve and $f^{\prime }(x)$ will be opposite to the sign of $A$ which is -ve in $\left[-\frac{1}{2},1\right]$ Hence, $f^{\prime }(x)$ is $+$ ve in $\left[-\frac{1}{2},1\right]$ $\therefore f(x)$ is increasing on $\left[-\frac{1}{2},1\right]$