Mathematics - Application of Derivative Question with Solution | TestHub

$h\left(x\right)=fog\left(x\right)=e^{g\left(x\right)}-g\left(x\right)$
$=e^{x^2-x}-\left(x^2-x\right)$
$h^{\text{'}}\left(x\right)=e^{x^2-x}\times \left(2x-1\right)-2x+1$
$=\left(2x-1\right)\left(e^{x^2-x}-1\right)$
$\because h\left(x\right)$is increasing$\therefore h^{\text{'}}\left(x\right)\ge 0$$\Rightarrow \left(2x-1\right)\left(e^{x^2-x}-1\right)\ge 0$
Case 1:$\left(2x-1\right)\ge 0$&$x^2-x\ge 0$
$\Rightarrow x\ge \frac{1}{2} \& x\in \left(-\infty , 0\right]\cup \left[1,\infty \right)\Rightarrow x\in \left[1, \infty \right)$
Case 2:$\left(2x-1\right)\le 0$and$x^2-x\le 0$
$x<\frac{1}{2}$and$x\in \left(0, 1\right)$$\Rightarrow x\in \left(0,\frac{1}{2}\right]$
$\therefore x\in \left[0,\frac{1}{2}\right]\cup [1,\infty )$