Question

Find the intervals of monotonicity of the following functions.
$f\left(x\right)=x.e^{x-x^2}$

Answer 1

$f\prime \left(x\right)=e^{x-x^2}+e^{x-x^2}\times (1-2x)\times xf\prime \left(x\right)\ge 0e^{x-x^2}[1+x-2x^2]\ge 0{2x}^2-x-1\le 0{2x}^2-2x+x-1\le 02x(x-1)+(x-1)\le 0(2x+1)(x-1)\le 0(x+\frac{1}{2})(x-1)\le 0x\in (-\frac{1}{2},1)x\in (-\infty ,-\frac{1}{2})U(1,\infty )$